My cold water pipe snapped behind the shut off valve on the threads inside the wall. I do not know how to weld properly and the main water supply for the whole house is shut off. What do I do?
Is it possible to bypass just one bathroom sink and allow water through out the rest of the house?

For starters, shut off the power or turn the gas to pilot on your hot water heater and open a hot water tap unless your hot water heater has an expansion tank.

How much of your piping is accessible and what kind is it? If you can get at the line after it tees off to the sink, you can cut it and cap it. Its copper, and you don't do sweat soldering? No problem, go to an Ace Hardware or plumbing supply and ask for a Shark's Tooth cap. Maybe buy a tubing cuter too. If the pipe is 7/8'' OD ask for a 3/4'', or 1/2'' for a 5/8'' one. Plumbers always talk ID, so you ask for a size smaller than the OD. The Shark's tooth works with any other type of pipe that has the same OD. If you have a white plastic pipe more like an inch, no problem. Go almost anywhere and buy a 3/4'' PVC cap and a can of PVC cement. Cut the line with hacksaw or anything, clean it up and cement the cap in place. Then post back, and we will help you decide if you want to tackle the broken pipe or not. The Shark's tooth cap is easily removed by pushing back a plastic collar. Leave yourself enough room before a PVC cap to saw it off and add a coupling.

Thank you labman for your response. It looks like three quarters of an inch copper threaded piping and it is broken flush at the wall behind the sink. A buddy suggested smashing out the tiles and cutting a big hole around the pipe but that sounds like it will lead to more problems. Is that what I have to do just to get a cap on it?

Copper pipe usually isn't threaded. A chromed brass pipe threaded in to a copper pipe elbow is common. Older work could be galvanized. Cutting and capping the line elsewhere only buys you time. Before busting the tile, try spraying penetrating oil in the pipe and working the end of the pipe over good with a hammer, and then an internal pipe wrench or a large screw extractor, Easy Out or other brand. Alternate, oil, hammer, twist. You could heat the pipe with a torch and then quench it by a quick shot of water. You may be lucky and have it screw out.

What is on the other side of the wall? It is easier to repair dry wall than tile.