One day it suddenly would not run for more than about a minute and then quite. Started up fine after priming; but, then soon stopped.
I figured, "Uh, oh; the carburator must have gotten gunked up because we were running it without the paper part of the filter. Tried cleaning the carburator-- no change.
Finally got a new carburator and a new filter. Didn't help. Maybe the gas cap vent was clogged up; so, tried drilling a hole in the cap to make sure it would feed gas to the engine; didn't matter. Drained out old oil and replaced it; same thing.
Then, after reading some stuff on the B&S site, it seemed like a good idea to check if the spark plug was wet or dry after the engine stopped. That way, when I asked for help, I could include that info.
So, I remove the spark plug and notice I'd somehow managed to break it during removal. Rats! (BTW, the plug did not seem especially wet or dry.)
Checking through our junk, there turned out to be a plug that was a bit longer but had the right threading and internal length. So, after matching the gap, in went the replacement plug.
I started the mower to make sure it would run the usual minute or so with the new plug. It ran fine. In fact, it kept running. The mower was fixed!
My guess is that the old plug had developed a fracture, which is why it broke so easily during removal. Evidently, after running for a little while, heat led to expansion and a separation in the conductive path inside the plug and it just stopped firing-- nicely mimmicking a fuel flow glitch and other assorted problems.
Maybe a fractured spark plug does not explain your 'engine stops running' mystery. But, if it does, finding the problem at the start can save money and a bunch of bother.