I too am having this same exact problem.
Mine is with my father's old lawn mower. I'm not sure if he had replaced the fuel line with an incorrect type of line, or if it was the original one - but either way, that line was totally turned to jelly. It was like black tar, it just pulled right apart in my hands. Needless to say, this didn't do much for the carburetor.
I took the carb apart and pulled as much as the black sludge out as I possibly could - slapped it all back together - still nothing.
I pulled it apart again and really gave it a good cleaning. I used a long, thin wire to clean out any tiny holes I could find - I pulled the primer bulb off and cleane out under that. I cleaned out the holes in the screw that holds the bowl to the carb.
I didn't bother with new gaskets or any new parts since I didn't have any.
I put it all back together (with a new fuel line of course) - and it ran perfectly. I cut their entire lawn with it which took a good hour at least. When I went to do the back lawn, it started failing again.
Same thing - runs for a second then dies.
I've taken the carb off again and tried cleaning it all out again, but I'm still having trouble with it. I believe new carbs are only about 40 dollars for these engines, so I'm thinking I may be better off just buying a whole new one and calling it a day.
I also noticed looking in the gas tank that there's all sorts of stuff floating around in the gas - it looks like some kind of stew. I'm sure this isn't helping matters. I'm sure some of that junk could definitely clog the carb since there isn't a filter in the line.
By the way - SHOULD there be an inline filter? I can picture my father leaving that out if he did in fact replace the fuel line at one time.
I may try to clean out the gas tank and give it all one more shot before I buy a new carb.
I'll do my best to get back here with an update - but there's a good chance I'll never bother - so I figured I'd throw my two cents out there now while I'm here. Cleaning out those little holes in the carb did help at one point. Maybe that can get somebody else on the right path.