See here:

http://www.hort.purdue.edu/ext/vinegar.html

The upshot is, vinegar's good for annual weeds but it won't kill perennial roots..although if you have to spray a little more often, that might be worth avoiding pesticides. (Remember, though, it will acidify the soil nearby.)

Also, store-bought vinegar generally isn't concentrated enough, so you need to buy a special herbicide-grade vinegar.

you can simply pour boiling water on the weeds.You will need to repeat it every 6 months or so because new ones will sprout again with time.You will kill all warms and whatever else is in the soil , but it works for me..

I want to second scarleta. We just tried the boiling water idea out last week and, while it did not kill every single weed outright, it did kill many and render others crispy and brown in the center. We will reapply when the weather warms up--according to what I've read, you need to do it on a very hot, sunny day.

Add lots of salt to the boiling water. The salt in the soil helps keep weeds from coming back. Since you aren't interested in growing anything there, that should work for you.