AMD has released AMD Catalyst 13.6 Beta drivers, with support for Ubuntu 13.04 Raring Ringtail and other Linux Systems.
The new AMD drivers have a lot of gaming and non-gaming new features. Oficially, the new drivers work on Red Hat Enterprise Linux suite 6.2 and 6.3, SUSE Linux Enterprise 10 SP4 and 11 SP2, OpenSUSE 11.4 and 12.1 and Ubuntu 13.04.
AMD Catalyst system requirements: Xorg/Xserver 7.4 and newer, Linux Kernel 2.6 and newer, glibc 2.2 or 2.3 and POSIX Shared Memory (/dev/shm).
The new drivers are not stable, so think twice before installing them.
This is how you install AMD Catalyst 13.6 Beta on Ubuntu 13.04 Raring Ringtail
This works for both 32 bit and 64 bit systems:
$ wget -c www2.ati.com/drivers/beta/amd-driver-installer-catalyst-13-6-beta-x86.x86_64.zip
$ unzip amd-driver-installer-catalyst-13-6-beta-x86.x86_64.zip
$ sudo chmod +x amd-driver-installer-catalyst-13-6-beta-x86.x86_64.run
$ sudo ./amd-driver-installer-catalyst-13-6-beta-x86.x86_64.run
Bonus, install AMD Catalyst 13.4 Stable on Ubuntu 13.04:
$ wget -c wget -c www2.ati.com/drivers/beta/www2.ati.com/drivers/linux/amd-catalyst-13.4-linux-x86.x86_64.zip
$ unzip amd-catalyst-13.4-linux-x86.x86_64.zip
$ sudo chmod +x amd-catalyst-13.4-linux-x86.x86_64.run
$ sudo ./amd-catalyst-13.4-linux-x86.x86_64.run