I bought one a couple years ago that has 512 Mb of storage but also had a built in FM tuner. I used the FM radio feature more than the MP3 player so the storage space wasn't an issue.
My new cell phone has an MP3 player and I have stopped using the small MP3 player and just use the phone now. It's an amazing phone too, it's a camera, video recorder, voice recorder, you can email, surf the internet, watch TV, Play MP3's, it has voice recognition and several other options I dont even know how to use yet.
There is a mini disk slot on the side so you can increase the memory for storage. I bought a 1 gigabyte card on eBay for $20.00 including shipping and now have more music on my phone than I have time to listen to in a day.
I would suggest if you are looking for am MP3 player with only 1 gig of space, try eBay first and look for the generic ones with the built in FM tuner. They work just as well as the Ipod's and are much less expensive.
Cheers,
Sean