It seems like a nice science project to go ahead and weld it, but the prudent thing to do would be to have it sleeved.
One more thing: I would bet an aggressive prosecutor could easily get the idea that taking a hacksaw to a shotgun barrel about 13 inches from the breech would fall under the heading of "attempting to make a sawed-off shotgun", which would make the saw operator eligible for an extended stay in the Graybar Hotel. The guy experimenting with welding up the old beater - what's he going to say? "Yeah, I was cutting it there, but I wasn't going to go all the way through?" As a pure scientific experiment, it might be useful to see whether barrels can be welded, but do consider the legalities which might apply. Before you wind up having to pony up $10k or more to ... a criminal defense attorney. (My brothers at the bar will be so pissed

at me, for giving advice that would ... cut into their business.

)
As to the mope working the hacksaw who started all this, from the looks of the barrel he was about as competent at operating a saw as he was at breaking, entering and getting away, i.e., not very.