Makes no sense. If the water supply was connected to the sink (I'm puzzled over this one, but I'll allow for someone to design such a sink with an integrated faucet), then there's no way in hell he's plugging the hole. Go turn the water on in your sink & see if you can plug the faucet with a piece of gum or something while the water's running. Not happening. Thus, the integrated water supply makes no sense. Water just flowing out of the sink also makes no sense. There is, of course, an overflow for sinks, but I can't see how that would..
ahhhh, your overflow had clogged & was full of water. If you try to run the water over the top of your bathroom sink, most have a little hole near the top for excess water to have an alternate path to the drain, rather than allow the water to overflow. You either have the wrong type of tailpiece for the sink, else you've been sticking gum or something into that little hole & now it's clogged & holding water from the sink previously being overfilled. Well, past tense - it WAS holding water. Regardless, sinks are cheap. Check to make sure you have the correct tailpiece when you install the new sink.
What the tailpiece should look like:
http://www.plumbingsupply.com/bathroom-sink-drains.html
Note the extra holes that would be located within the thickness of the sink for overflow water to exit through.
What you might have:
http://images.orgill.com/200x200/6107460.jpg
You might be able to patch the hole, but it's going to be difficult to make it look right. I'd try some sort of epoxy, if you can find an epoxy that matches your sink color. Do it in two stages - first insert a piece of cardboard through the hole, with a piece of string attached to keep it pulled up against the sink & act as a backer. Then, with the string held firm (or tied tight to something), fill the hole half way with epoxy. After it dries, clip the string & fill the remainder of the hole with epoxy.
Alternately, a little googling shows that there are porcelain repair kits for sinks & the like; but matching the color perfectly still might be tough. It's cheaper to go that route though.