Vinegar. Here's a lot of good information and opinions about vinegar for weeds:
Putting down a good quality week block cloth with a covering of hay should do the trick. You will get a few coming up, but if you pick them out when their small it is easy to control them.
Katrina H, you can do it inexpensively if you just put down newspaper & or cardboard, soak it and cover with mulch of any kind.
I agree with Sherrie S - it's inexpensive and effective! No chemicals - and will compost in the garden later.....make sure you layer thick.
Found this on Pinterest- uses vinegar, dish soap and salt- Here is the link-http://www.creativehomemaking.com/garden/homemade-weed-killer.shtml
I always lay a good layer of newspaper down, wet it, and then put my grass clippings over it all summer long. I even get our neighbors clippings. The weeds that do come up are easy to pull. The bonus is we don't have to till each spring, just turn it over, it's extremely easy to dig. And... we have earthworms galore!