I've visited the Black Burn, and apart from a very conspicuous (but probably natural) mound and lots of muddy water, there was indeed nothing to be found. Pauperhaugh is close to Longfamlington, and the findspot reported in Keys To The Past matches up with Pebbles' description above.
This stone was brought into the Durham Rock Art Conference on 8-3-2003 in the back of someone's car. The delegate had found it in a stream near Longframlington, and was now in a carrier bag, so there's no point in trying to find this one :-) Stan Beckensall got very excited about it.
One side is green and more weathered. The other side is very well-preserved, with clear pick marks. There is a large chunk missing.
Comments were made on the way the green side used a natural line in the stone to look like a face ?!