I am installing an over the range microwave. I am adding a new dedicated circuit to plug it into. My problem is this. The old range hood was direct wired. There is a three wire romex cable coming thru the wall. This is connected to all the other ceiling lights in the house. There is no junction box. How do I deal with it? Can in install a box where the wires come through, and just leave the live wires, capped inside? It would be behind the new oven. There is no way to access the wire other than to open the wall completely which is not an option since there are already cabinets installed.
How do you know that the wire is on the same circuit as the lights??
I can clearly see the wall so you can open it.
If you are not opening the wall how are you installing a dedicated circuit for the new microwave and where are you putting the outlet for that?
"Can in install a box where the wires come through"
YES. You can install a "old work" box there, (made expressly for retrofit). Cut a hole around the wire carefully just big enough for the box, use the box as a template and trace around it with a pencil. Run the wires in to the back of the box and cap them with wire nuts,insert the old work box and tighten the screws that pull the tabs which old the box to the drywall, put a solid cover on the box.