I believe the pipe has a very tiny crack where the water is coming from. Trying not to call a plumber if not necessary.
First see if it is a leaky valve or leaky solder welds. If it is the welds then you can heat it up and try to resolder it. If it is the valve you will have to replace it. You will need an identical valve, some solder, a propane torch some flux, and a wire brush. Make sure you shut the main valve off and turn on some faucets in a lower level if possible to empty the water out of the line or you will boil it in the line. heat up the joint with the torch until you can pull the valve off the line. Pull it straight off both sides then don't touch the line. Take your new valve (make sure you get the same one a sweat type valve) brush out the inside real good on both sides with a wire brush then put some flux on it. Then you wanna take your torch and torch the pipe with the solder on it and the valve until the solder melts and you can slide the valve back on the line. Put a little extra solder on there in case the fit is not perfect or you lost a little during removal. All you need to do is touch the joint with the solder, the flux will pull the solder into the joint and fill it. Do the same with the other side. Wait a few minutes for the solder to cool off then try turning the water back on to see if the leak is stopped. If it is, well done. If its not, try again or call a plumber.
The copper pipe leading downward into the shut off valve under the sink is corroded. Will this cause a flood eventually.
It sounds like it it the pipe between the sink and the valve that is bad and that's good! There are two possibilities. ---If it is a bent copper tube attached by a nut to the valve: Measure it's length and note the size of the nut where it is connected to the valve. Then go to you local hardware store or home center and tell them you want a braided supply line for a sink that is at least as long as you need. The sink end is standard and there is only a few possibilities for the valve end. For the Valve end you should be able to tell by eye what size it is. --- If it is a 1/2 inch copper pipe soldered directly into the valve: go to you local hardware store or home center and tell them you want a Shark bite Slip Coupling for 1/2 inch pipe. (and a pipe cuter if you don't own one.) Cut out the bad section, clean the ends and reconnect using the shark bite fitting (no tools required.)
