>Mike,
>
>The tank valve has 2 seals - one for "open" and one for "closed".
>
>Since your leak stopped when you closed that valve, I suspect you have a
>faulty seal on the valve, not the regulator. Not an expensive repair, but
>also one that I would not do myself.
>
>Propane fumes are very flammable. Let someone who does work on propane
>tanks take care of this for you, and stop using the propane until you get
>the tank repaired.
>
>Michael

I did the soap bubble test, and it turns out the leak is coming from
a tiny pinhole on the bottom of the regulator.  It's hard to tell
whether the hole is part of the unit, a manufacturing defect or
perhaps a rust/puncture, but in any case I've decided to replace it.

My original concern was that perhaps the regulator was built to bleed
off a tiny bit of gas now and then as part of its normal operation,
but the amount coming out is not acceptable.  So either it's got a
hole in it, or it's malfunctioning, same conclusion : replace it.