Samsung might not release the official Jelly Bean update for the Galaxy S3 in the near future, but custom the ROM development scene shows good progress.  CyanogenMod is still the JellyBean ROM which is compatible with the greatest number of devices, but several other Jelly Bean ROMs like LiquidSmooth and Resurrection Remix are now available.

In order to install CyanogenMod 10, your device should already run on a custom AOSP-based ROM like CyanogenMod 9.  Before the installation, make sure that your device is at least 80 percent charged. Back up apps and important data like contacts and messages.  

Please note that this build is only compatible with the international version of the Samsung Galaxy S3 (model number i9300). You may "hard brick" your device while installing this ROM on the LTE variant of Galaxy S3. We do not hold any liability for problems, errors, or damage during installation, even with the international model. 

Here is the step-by-step guide to install CM10 Jelly Bean on the Samsung Galaxy S3.

• Download CM10 Preview#10 and Gapps package to your computer.
• Copy the downloaded files to phone's internal SD card memory.
• Reboot your phone in ClockworkMod Recovery mode.
• Select 'Install zip from SD card' and choose the CM10 package that you placed on phone's memory earlier. Confirm the action and wait until the process is over.
• Install Gapps package, which includes Google apps like Gmail and Play Store, using the same process.
• Go back and select Wipe data/factory reset (this is why it is important to back up your data before installation).
• Reboot your smartphone into the new Jelly Bean operating system.