Currently the stable version is version 2.3.5, and I wanted to go with the 3.0.1 experimental version, and here’s how to do just that:
1. Go to the Archidroid’s github repository: i9300-cm-experimental
2. And download the repository zip: direct download zip of i9300-cm-experimental
3. After the download is finished, unzip the archive.
4. Download 7zip from here: 7zip.org
5. After you’ve extracted the i9300-cm-experimental zip archive, go into the directory, select everything (Ctrl+A shortcut) and right click and select 7zip > Add to archive… and select Compression Level: Fastest and the Compression method: Deflate and click ok.
6. After the zip archive has been created, copy it to your device.
7. Next, just reboot your GS3 into recovery console and flash the .zip archive like any other rom.
A short video available on the Archidroid thread which shows some of the steps mentioned above:

It doesn’t require anything else, and it will grant the user full root privileges without asking and without using any special permissions.
Even though, this is not something you should take lightly (as the hack could be used by third-party apps with malicious purposes), it is the fastest, simplest way to date, to get root access on a Samsung Exynos4 type device running factory stock roms. (Finally a simple method of using Titanium Backup without flashing custom images, bootloaders, unlocking device, etc.)

The current version of the app is 1.20 and starting with version1.10, it has the options to disable the exploit (disclaimer: this may break your camera), re-enable the exploit (if you need the camera) and to disable the exploit at boot (before any Android app runs). These options do require root (SuperSU or Superuser) to be installed as well. While this will help protect you, these are workarounds, not actual fixes.
All you need to do is download the app from the links provided below (or the original forum threads), install it (make sure you have enabled “installing from unknown sources”), and click the “Root device” button.
Samsung has been informed of this vulnerability, and it is likely that a fix will be available in the near future, but until then, if you were looking for a 1 click ROOT your Galaxy S3 procedure, this is it.
The method has been tested and works on the following devices (check the forum thread below, as it will be updated with new devices):
Notes:
Download: ExynosAbuse-v1.20.apk
QR Code (scan code to download app):
Sources – XDA developers forum threads: