First of all, smaller, thicker jump rings are stronger than larger, thinner ones - simple physics at work here! If small, thick jump rings aren't strong enough then you can either solder, use split rings or Snapeeze rings.
If you choose to solder, you'll have to consider the metal you're using and choose your soldering tools and equipment accordingly. Base metal has a nasty habit of reducing to a little molten puddle when you take a torch to it. If you're working with sterling silver, you'll need approximately $75 of equipment for a very basic set up that includes a small butane torch. This is what I have and it's plenty good enough for small soldering jobs, such as jump rings. If you get into playing with fire though (and it IS a lot of fun!), you'll want/need a more powerful torch. Here's a link to a great "how to" video: http://www.youtube.com/watch?v=nYVl35kvKHs.
Split rings are like a keyring - http://www.worthco.com/rings/spltring.html (I've never ordered from this co, this link just shows a pretty good picture of a split ring). Personally, I don't like them - I think it makes jewellery look amateurish and "home made" but hey, that's just my opinion for what that's worth!
Your third option is Snapeez rings. I've never used them but others on this board have (if you try doing an advanced search and set the date back at least a year, you'll find more info.). Snapeez are available at http://www.viamurano.com/shopping/ (BTW, I couldn't get the Snapeez "how to" video to work????)