Cut the Clematis back and dig it out. You can cut the Clematis way back.

If you're talking about the same plant that I call bindweed (elongated leaves, little pink morning glory flowers) then you won't be able to kill the bindweed by digging it out. Its easily breakable roots go down 10 feet or more, and every spot you break the root, it sends up new top growth.

How entwined are they? You really have two options; you can cut the bindweed at the base (at ground level, without disturbing the roots) and pull it out of the clematis (or wait till the bindweed vine dies, and then break it out of the clematis, if pulling it out would damage your clem). You will have to do this each time the bindweed starts to regrow, as this technique is a cosmetic fix. However, snipping just the top growth and not disturbing the roots makes it grow back much more slowly.

Your other option is systemic poison. You would need to unwind as much bindweed as you could from the clematis, and then dunk it in a solution of 2-4D (a systemic broadleaf weed killer). The bindweed absorbs this through its leaves and transports it to the roots, and all the top growth will die in a few days. You may have to reapply in a few weeks, if it starts to regrow. However, this is a risky proposition; the poison can also damage your clem if you get any onto its leaves (it will damage any dicot plant; grasses aren't bothered by it). You would need to dip the bindweed in the poison (don't spray, as you'll get drift), and then keep it away from your clem while it dries.

Unless you have a rampant weed, I'd just go with the chop it off at ground level approach. This is quite manageable, especially with a good layer of mulch over the decapitated weed!

Good luck!

Lara