It looks like a weed in the morning glory family of weeds, which includes bindweed--did you notice any flower in that area last year? Anything trumpet-shaped perhaps, in white or pink/purple?

Delicate and vining certainly sounds like bindweed/morning glory. It can be horribly invasive. I am battling some on a steep bank, trying to cut it down and mulch it under.

No the vine doesn't have any flowers - I don't think it's from the morning glory family - the leaves are quite small and vine stems very thin but tightly wound around...they break since they are so delicate when you try to pull them out if you find one before it has a chance to wrap around a plant. They're growing in full force right now - even wrapping around lily leaves and astilbe - wherever they can. :-(

Just pull them from the base as you see them--it's aggravating and time-consuming, but they're not strong so as long as the area that you are trying to keep them out of isn't too large, then you have a decent chance of stopping them.

They won't have any flower yet--but if they're from the morning glory family they'll have a tubular flower later on. Don't let them flower--otherwise they produce seeds for future years!

I pull the stem out of the ground and often just leave the twining vine on my plants. Eventually it will dry up and break.

I agree with what's been said already. Bindweed/buckwheat weed of some type. The difficulty in trying to identify exactly which weed vine this is, is that there are so many, each with several common names, AND their leaves can be variable depending on who the parents were last year, what part of the country you are in, its' stage of maturity. It's almost as if they are able to disguise themselves to fool people. Some make morning-glory-type flowers, and some make clusters of little white or even greenish flowers that most people never notice. Identifying it is secondary to getting rid of it.

If you go to this discussion and read the info posted by Bry84, the 4th entry, you will find good instructions. Good luck!