You'll find a lot of opinions/articles on sound post position/adjustment.

Adjustment is best done with a focus on the overall sound the instrument and all of the strings rather than a focus on changing the tone of one string...

So any movement from its general position should be minimal.

I bet there's a better way to address this, but seeing as I take my fiddle to a very talented luthier, I can't offer any other advice than, don't mess with it without a LOT of research. Asking here isn't really doing research  or is just step one. 

Prim does make an "orchestra" (heavy) weight set of strings, and I imagine the A string is available individually. Beyond that, I'd consult a luthier.

I'll add this: the reason I say "very talented luthier" is because my man Randy Hoshaw improved every fiddle I've taken to him, each of which had been set up by a variety of luthiers from violin shops (and music stores and hobby tinkerers). Whatever he does, he seems to know the instrument's specific idiosyncrasies better than most. And since I'm singing his praises - he is also a painter, potter, caretaker of bonsai and maker of unique and visually pleasing furniture.

Why not have someone else play your fiddle while you listen from across the room, see if you still notice the same difference. There are lots of potential reasons for what you describe, and its a problem best addressed by someone with experience and skill setting up fiddles.
Unless you have the skill set, you could do more harm than good, just my two cents.

Higher tension strings don’t necessarily produce more volume. Dominants are some of the lowest tension strings on the market, and they are regularly used by soloists in the largest concert halls in the world and can be heard over a full orchestra. Higher tension changes the kind of sound the violin makes and it may promote a response at different frequencies.

I would not recommend mixing the tensions on strings, though. Sets are carefully engineered to work together. Changing the E string is standard practice, but E strings are usually single strand steel and the differences, although noticeable to the ear, are more structurally subtle.

If you’re noticing an imbalance in sound, the cause is most likely a soundpost issue which may come from placement, tension, or fit. It may seem simple to just tap it around with a soundpost setter until it sounds the way you want, but doing so poses several risks:

1) You might knock the post down. That is dangerous for a violin under string tension because it can cause cracks. If you don't know where it belongs, trying to reset it will be an almost impossible task. It needs to be oriented along several axes, and you can't get a sense for its orientation with the violin fully set up unless you're very good with an inspection mirror and know a few tricks to check orientation from outside; even then, you're unable to be as certain as you'd be with the setup down and looking from the endbutton hole. 
2) If you’re moving a soundpost around a lot, there’s a good chance you’ll damage the top. Especially if you’re moving the post in a way that adds tension, the endgrain of the soundpost can really dig into the top. This is how a lot of soundpost cracks start.
3) If you don’t have experience with a soundpost setter, you can do a lot of damage to the f holes very quickly.
4) If you move the soundpost to another spot, the tension and fit may also change, and depending on how far you move it, that change could be significant. A soundpost needs to have proper placement, fit, and tension to function at its best. Adjustment is really about making movements of fractions of a millimeter with a post that meets the three criteria. Once you’re moving more, you risk losing at least one of those things essential things, and you need to know how to assess them.

A lot of people can cut a passable bridge and soundpost, but there are not many who know how to adjust instruments well. That requires an intuitive understanding of violins and a very good ear to understand the way the instrument responds, both under the ear and at a distance.

Is finding a good luthier so out of reach? I know they charge $, but they also save $ (because mistakes are expensive) and also a huge amount of heartache. Also, I notice some folks are sort of addicted to do-it-yourself-ism, but really, it's okay to consult a professional. You'll likely be amazed at what they can do.

Excellent and thoughtful comments, thanks. I think I'll start by having a friend play the fiddle while I listen from across the room. Maybe there is less unevenness across the strings than I sense with the fiddle under my ear. 

Follow up: I invited a friend to play my fiddle while I listened from across the room. I could not detect any significant difference in the volume of the strings. All four strings sounded equally strong. Two other listeners agreed.

I was careful not to let the others know what I was looking for, so I'm reasonably confident that my weak A string is something that is perceptible only at a distance of about six inches. I have no idea how that works, but I guess that's just another one of those Mysteries of Life.

Thanks for good advice!

THe under-the-ear sound of your fiddle is where even the most subtle differences are most clearly heard. Me personally, if I am not happy with the U-T-E sound of a fiddle I am not going to be happy and would either start having things changed/adjusted or just sell the fiddle and move on. I have a 1874 German/Tyrolian fiddle that to me sounds bad unless it is played for a day or two. Other folks say different. I keep it for sentimental reasons.

I think you have to pay attention to the sound at a distance just as much as that under the ear. Although it’s great to have an instrument that sounds consistent from in both places, many great instruments are known for sounding very different under the ear.

One of my customers played the Lady Blunt Strad when it was exhibited a few years back. He was shocked by how metallic he thought it sounded under his ear, but once you were about 10 feet away it became this amazingly beautiful and rich sound that was incredible to behold and there was no trace of the harshness.

One of the greatest violins I’ve ever played sounded glorious both under the ear and at a distance, but it didn’t sound huge under the ear. I would aways worry about whether it was carrying well enough when I played it in large rooms or halls so my father would go to the back of the venue during rehearsals to check the sound. Every time he said it was incredible how much definition in the sound came all the way to the back. Over time I learned to trust the violin and never doubt its carrying ability.

Many players like to test a violin in multiple settings before choosing. Joshua Bell famously fell in love with his current Strad at a shop and asked to take it with him for a concert that evening to be sure about the purchase. That performance sealed the deal for him.

As you see more and more violins you develop the ability to judge them quickly and to make predictions about their characters, but most players only play a few instruments during their playing careers, so they aren’t always able to make a decision on the spot.

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Originally posted by The Violin Beautiful

I think you have to pay attention to the sound at a distance just as much as that under the ear. Although it’s great to have an instrument that sounds consistent from in both places, many great instruments are known for sounding very different under the ear.

One of my customers played the Lady Blunt Strad when it was exhibited a few years back. He was shocked by how metallic he thought it sounded under his ear, but once you were about 10 feet away it became this amazingly beautiful and rich sound that was incredible to behold and there was no trace of the harshness.

One of the greatest violins I’ve ever played sounded glorious both under the ear and at a distance, but it didn’t sound huge under the ear. I would aways worry about whether it was carrying well enough when I played it in large rooms or halls so my father would go to the back of the venue during rehearsals to check the sound. Every time he said it was incredible how much definition in the sound came all the way to the back. Over time I learned to trust the violin and never doubt its carrying ability.

Many players like to test a violin in multiple settings before choosing. Joshua Bell famously fell in love with his current Strad at a shop and asked to take it with him for a concert that evening to be sure about the purchase. That performance sealed the deal for him.

As you see more and more violins you develop the ability to judge them quickly and to make predictions about their characters, but most players only play a few instruments during their playing careers, so they aren’t always able to make a decision on the spot.

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This is my problem. I think my fiddle or even my playing sounds "decent" however, the recordings always prove differently lol.

I like to hear others play mine but it's so deceiving the difference between under the ear and from a distance. 

Erockin - recordings have so many different variables than live sound though. I don’t know if they have more variables, just different. They give you one perspective, but it’s still not the whole story.

consider this: if you move the post laterally, that is, to or from a C bout, it will be either too loose or too tight..Don't do that.. If you move it between neck and tail, it will become either too loose or too tight. Don't do that either. In other words.. unless you know exactly what you are doing... do nothing..(I'm speaking from years of  in-experience)

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Originally posted by Strabo

I have a fiddle with generally excellent tone, but the A string seems weaker than the others. I currently use Prim medium tension strings but I have noticed the weak A with other types of strings. 

I looked for a heavy tension A string, thinking that higher tension should produce greater volume. The only heavy A tI could find was Helicore.(I have used Helicores in the past and I like them but they do not speak as  clearly as the Prims.)  Anyway, the heavy Helicore A didn't make much difference.

I have heard that soundpost position has a significant effect on tone, so now I'm thinking about adjusting the post. I have the tool but I have no idea which way to move the post -- or even if it is reasonable to expect such a specific change of tone or volume. 

TBH, I don't really know that the weak A is perceptible to listeners -- maybe I'm the only person who notices it. But I am wondering if I can make this good fiddle a little better.

Or maybe I should just accept the current situation and play the A with a little  more juice.

Any ideas?

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Try moving the sound post back away from the bridge. This should increase the volume of the treble side. 
      Before you move it , draw a picture of where it is now , just in case.
        Also, before you move the post, make sure that the feet of the bridge fit well. It is possible that you are losing amplitude here.