Well, good luck in finding it.
No site that has a complete listing of Apple's hardware (there are a few more tha just apple-history), neither Apple themselve, list a B&W G3 All-In-One as existing model.
The beige AIO was produced only for a couple of months (April '98 until January '99), and was meant only for the educational market. So I highly doubt they made a limited modell within that already limited time for a limited market. It has been replaced by the first, Bondy-blue iMac.
Even the hardware was that from the beige G3, not of the B&W. Because of a limitation in the beige's hardware, you need a partition of less than 8GB to be able to install OS X. The B&W doesn't have that limitation anymore.
But it really would be very interesting if you can find a pic of a B&W AIO (NOT Photoshoped, of course!!)...because it would show that Apple has some very interesting rarities noone knows about.