From Where you can access iOS 6.1.3 Jailbreak then you have to read furthur more about it.
Apple iOS 6.1.3 Apps and Jailbreak Official Website : http://www.howtojb.com/
Apple Software Update iOS 6.1.3 Compatibility:
iPhone 4GSM Model
iPod Touch 3rd generation
iPod Touch 4th generation
The following steps will help you jailbreak your device on Apple iOS 6.1.3
Step 1: Go to apple.com and Download latest iTune v 10.2.1 and Update your device to Software Update 6.1.3
Step 2: Then go to http://www.howtojb.com/
and Download iOS 6.1.3 Jailbreak for Mac or Windows
Step 3: Save the iOS 6.1.3 Jailbreak Tool on desktop and Run as Administration
Step 4: Select your device and click on jailbreak button.
Step 5: Let the application do its work and wait for the app to show "Jailbreak Complete!" message.
Step 6.1.3: At this point, wait for the device to automatically reboot otherwise Reboot it according to the messesge.
Step 7: Finish! You should now have a fully jailbroken iPhone & iPad & iPad firmware iOS 6.1.3
Jailbreak iOS 6.1.3
Jailbreak Firmware 6.1.3
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