Looking at the ubuntu system requirements page, they say: "From experience, we all know that it is recommended to have 2048 MiB RAM to properly run a day to day Ubuntu." .
Of course you can run something else, but for a hassle free experience , ubuntu seems like the best choice.
With Unity you certainly need more RAM but nothing prevents one from installing another wm like xfce, awesome or lxde to save a ton on RAM and run just fine on 1 Gb.
Of course you can run something else, but for a hassle free experience , ubuntu seems like the best choice.