The worst thing with bindweed is that it has an underground creepingroot system that act as a storage organ and unless every bit of the roots are removed it will spring up year after year.
If you decide to use a weed killer, the best type is a Systemic weed killer such as 'Glyphosate' this will translocate through the sap stream of the plant and kill it right down to the root. ('Round' up is a popular choice) Great care must be taken to avoid getting the herbicide on other wanted plants. I would suggest buying the ready mixed spray gun or painting the leaves with a Glyphosate gel. Reapete applications may be neccessary with this difficult weed and it will be more effective when the weed is activly growing in spring and summer but even now would have good results and you can then see what comes up in the spring. Good Luck Tbird+
There was a similar question to this in Gardens question time on the radio this week. In this case it was coming from a neigbouring garden. They suggested putting in some deep slabs on the boundary to prevent it coming in. As yours is rampant I suggest you do this to yours if you wish to remain friendly with your neighbour.
Just like to say, combine the answers from T.bird and spudqueen, its the method Kew gardens uses, found that out on a visit there, I used this method, a cane and Glyphosate, and it worked.