I am looking for a smal digital alarm clock with full brightness
control - that is the ability to have it completely dark and invisible
and increment it up in brightness from there, to where it is just
barely visible in a completely dark room.

Nearly all alarm clocks I have seen recently have a button for
brightness control that allows choosing 2-3 settings, but alas, even
the darkest setting is too bright for me in a very dark room. I need a
clock that can be set to nearly fully invisible, perhaps a dial control
allowing full flexibility with numerous possibilities instead of a
button-selector with 2-3 permutations of brightness, all of which are
invariably too bright. Please include link to details on it, with
brightness selectivity, model # and link to where I can buy said
clock.

Thanks!

Have you tried one of those Oregon Scientific Projection Alarm Clocks? 

They are completely dark, as in you can't see any glow on the clock
itself. The clock then projects the time onto the ceiling above, in
red characters. There is no glow or brightness at all. Of course it
doesnt have a Brightness control, but I had the same issue with clocks
that dreamed of being beacons and this was a perfect alternative. The
new ones even come with a radio, meaning you dont have to be jarred by
that hellish beeping.

 - artqs

www.oregonscientific.com

I have found clocks that suits your needs perfectly!

Cheap solution: Sonic Shaker SBP100

It only lights when you press a button; otherwise it stays completely
dark. To find out where you get this alarm clock:
http://www.sonicalert.com/htm/state.htm

Expensive Solution: Flemming Bo Hansen: Digital Alarm Clock 3201 
Buy it at: http://www.questodesign.com/shop/proddetail.php?prod=rosendahl_3201
    Product description (from Amazon): Product Description
Made in Denmark. A wonderful modern clock that does not keep you await
at night with a harsh "thicking noise". High functionality is at the
core of our new series of timepieces. The alarm clock has a digital
display and provides a very soft lit display. Choice of between a
12/24 hour display. The alarm clock has an alarm and a snooze function
and it lights up automatically when the alarm is set. The alarm clock
is made of black polypropylene and steel and is designed by Flemming
Bo Hansen.

I appreciate the comments guys, and I checked the links - but those
don't meet my needs. The projector is still too bright. At one point
years ago I had a clock with a dial to select brightness, and it shut
all the way off, this was perfect, but I can't find it.

Most of the backlit clocks are too bright on the lowest setting. The
suggestion about the clock where you hit a button to see it is, good,
but I don't want to go through that trouble. I want a really really
really dim digital alarm clock.

Please keep looking, you will be rewarded! 

Thanks.