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You can jailbreak an iPhone 3GS on iOS 6.1.3, but only a tethered jailbreak.

An old bootrom 3GS will always be untethered

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Trying to JB a 3GS (Old Bootrom) on FW 6.1.3 with redsn0w 0.9.15b3

Supposedly by running off the 6.0 ISPW you can JB

I get the message : "The connected device does not support untethered options"


http://www.iphonehacks.com/2013/03/jailbreak-ios-6-1-3-using-redsn0w.html

How are you trying to jailbreak the device? A list of steps you've done?
 
If the serial number is XX117, then it was made in 2011 and is a new bootrom. Only 3Gs' made in 2009 before November are old bootrom.

Your right, I was thinking of XX134 or later on the baseband specs. JB took after confirming it was a new bootrom after selecting 6.0 ISPW.
 
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