Hi Merryl and welcome to GoY. I'm sorry but this sounds like blight to me, can you take a photo and add it to your question. Copy and paste the link below into your web browser:

http://bit.ly/maoUyf

If looking at the info you think it is the only thing to do is to remove the potato plants completely including the tubers and destroy them. Don't put them in your compost and don't grow potatoes or other solanum, for example tomatoes, in that area of your garden for at least four years.

Sorry to disagree, if it is blight then you do not need to dig up the tubers, all you need to do is to remove the foliage. The spores of the Blight go from the soil up to the leaves, grow and then are washed back down to the soil and into the tubers to continue the cycle. As long as you get the blighted foliage (if it is Blight) off before the spores drop then the tubers, though small are still edible.
If this had been known in Ireland in the Potato Famine, then at least there would have been some potatoes to eat. They do not usually store well though.

If it is blight and you do decide to let the potatoes remain in the ground to use check them very carefully before cooking.