First: Make sure that your device is a Samsung GT-i9300 and not one of the many carrier specific devices on the market
To check tap 'Menu'
Choose 'About Device'
You will see an option marked ‘Model Number'
If it doesn't say ‘GT-i9300' or includes supplementary numbers or letters you should do some further research, as this guide - which comes courtesy of the users over at XDA Developers Forum - may not work for you
Next, before we go into things any deeper, you should make sure you have version 1.85 of ODIN, Samsung's smartphone software management tool - it can be downloaded from here (forum.xda-developers.com/attachment.php?attachmentid=581913&d=1304094576)
Once the download has completed, install ODIN
You should also grab the CF-Root-SGS3-v6.0 package - it can be grabbed here (http://download.chainfire.eu/185/CF-Root/SGS3/CF-Root-SGS3-v6.0.zip)
After your download completes unzip the file and place the resulting .tar file somewhere memorable, we recommend your desktop
Do not plug your Samsung Galaxy S3 into your computer yet
Click the button marked ‘PDA'
Choose the .tar file you unzipped earlier
Next, we're going to need to put your Galaxy S3 into download mode, and we this by switching the device off, waiting 10 seconds and then switching it back on again whilst simultaneously holding down 'Home', ‘Volume Down' and the ‘Power' key
Once you've done that, plug your device into your computer
Before you continue, make sure ‘Repartition' isn't ticked
Wait a moment
Ensure that you do not unplug your device as this may result in damage
Once the procedure is complete your device will be rooted