On ubuntu 16.04 you can use official apt package without install any other ppa repository.
sudo apt install rustc
... and don't forget
sudo apt install cargo
... but the versions are not really updated: (August 2016)
rustc 1.7.0 and
cargo 0.8.0. Unfortunally cargo is not compatible with IDEA rust plugin... I used the script pointed in Rust Documentation.
Jonathon Fernyhough has a PPA (personal package archive) where he provides unofficial nightly and versioned builds of rust, but it does require libstdc++ 6.x (Xenial uses 5.4.0). With Yakkety you can install Rust 1.10 from the universe repository, and 1.13 with Zesty.
Rust can be installed from this PPA by running the following, as well as llvm which is now needed:
sudo add-apt-repository ppa:jonathonf/rustlang (accept to add llvm) sudo apt-get update sudo apt-get install rustc
Alternatively different backport ppas like
rustlang-test can be substituted instead of
rustlang to get a the latest unstable code or a particular version.
I see there is a Debian prospective package bug 689207 still open. There are initial packages but it's not ready to be in the distribution. There is also a Debian wiki page about the packaging effort.
It alludes to the fact that Rust's compiler is written in Rust so the bootstrapping process is strange, so perhaps that's why it's not packaged yet.
There are some issues in upstream Rust that make it hard to package. Bootstrapping is apparently not a catastrophic problem as packagers can start from a binary snapshot.
Updated May 2016: Happily, rustc is now in Debian testing, so should be in Ubuntu within a year or so.
I might be too late in commenting here but I found that using https://rustup.rs/ was easier. Here is what I ran in terminal to have rust installed on my Ubuntu subsystem for Windows 10:
curl https://sh.rustup.rs -sSf | sh