Freebies and lightweight packages like Microsoft Movie Maker and Premiere Elements perhaps?
Premiere Elements needs 2GB RAM, Movie Maker - 1GB, will run on 32-bit XP systems. There're also similar lightweight editors from Sony, Pinnacle Systems. Look at their specs, see if they're OK with 1GB RAM and try them out.
You could also look at older versions of popular software - older than 5 years, perhaps even 10, but watch out for HD support and codec compatibility. From what I remember, Vegas Pro and Edius were relatively lightweight.
Amith, if you mean that 1 GB of RAM is the total of what you have available, I think you are out of luck. As Alex points out, 2 GB is generally the minimum for the application itself -- and your computer likely needs another 1 GB just to operate.
I just took a look at the solutions Alex recommended, and they all want at least 2, preferably 4 gigs.
Here's the exception: if you're working with small size files that don't require much processing. This rules out anything HD, and probably anything MPEG, which requires lots of processing power to encode and decode.
Since Creative COW is oriented toward professional editing, where upwards of 100 GB of RAM is not uncommon, we may not be able to offer much more help than this.