My stopcock won't budge
You should probably avoid trying to shut off the stuck stopcock by applying large amounts of force either from some type of lever or large adjustable pliers. The danger is that you may break the stopcock in the process and end up with water spraying everywhere. 

The best way to look at this is to turn off the water supply at the main cutoff valve at the street or entrance to the building. Then replace the stuck valve with a new one that works and adapts to your new functionality. It is a fact that valves like this are rarely used and builders use the cheapest units available. 

If it not convenient to replace the whole valve sometimes you can just use the main water shutoff so you can adapt the plumbing down stream from the stuck valve to get the dishwasher installed. But then make sure to put a note in your to-do file to get that stuck valve replaced as soon as feasible.