I am remodeling my kitchen.  The existing cabinets are in good
condition and are not going to be replaced.  However, the cabinets are
older and do not have enough depth (wall to front of cabinet) to
contain a built-in dishwasher made to today's typical specs.  I have
found today's dishwashers to have a depth of approximately 24".  The
specs for height and width of the dishwasher I am looking for are
flexible.

Specifically my question is.....Please provide me the makes and models
of dishwashers that have a depth of approximately 19-20" subject to
the following conditions.....no counter-top or portable models,
built-ins only....the product must be readily available for shipping
from a North American retailer and must cost less than $500.

Respectfully, I have searched and searched and I gather so have at
least a few GA researchers, (as I am not one), and your request is
nigh impossible. Even as I say this, somebody might prove me wrong,
but the research level is more time consuming than $5 would cover.
This project requires significant lookup of each possible dishwasher's
specs, which can mean up to 5 mouse clicks per model.

As you may have seen, there are at least a few 18" WIDE dishwashers,
but the narrowest depth I've been able to see have been slightly less
than 23 inches, which doesn't fulfill your request. There really
aren't that many dishwasher brands, and fewer models still within
those brands, most of which use the same case and change the innards
and the color.

If I might make a suggestion, I'd ask how likely it would be to remove
the wall behind the cabinet/dishwasher enclosure, or resurface your
cabinetry, or consider an 18" dishwasher oriented "deep-wise" -- that
is, rotate it right or left instead of front.

This is a free comment.

Thanks for your comment; I have continued my search as well, but to no
avail.  It looks like I am going to do some creative carpentery to
make this work to make it look good and retain the old cabinets. 
Thanks for your time.

There is always a solution, and I hope this is yours:
http://www.countertopdishwasher.com

These dishwashers will fit where no others will.  They are not toys,
they are real units with their own pumps, filters, heaters. The Haier
units are the best quality, I own one.