Devices that can be jailbreaked using this method:
iPod touch 4G
Back up iphone with iTunes.
Redsn0w 0.9.15b3 [Windows / Mac] A4 device running iOS 6.1.3 final.
iOS 6.0 IPSW.
How to jailbreak iOS 6.1.3:
This is a tethered jailbreak and will require you to plug your device into your computer in case you wish to reboot your device.
Step 1: Download the original iOS 6.0.
Step 2: Launch latest Redsn0w 0.9.15b3.
Step 3: Click on the Extras button and then Select unmodified iOS 6.0 IPSW.
Step 4: Go back to main screen of Redsn0w and click on Jailbreak and follow the on-screen instructions. Install Cydia when prompted.
Step 5: On completion of the process, enter DFU mode with the help of the Redsn0w wizard.
Step 6: Once the jailbreaking is done with the Limera1n exploit open redsn0w, then Extras and click Just boot to boot your phone tethered. Every time you need to reboot you will need to follow this step since it is a tethered jailbreak.