Ah, OK, I should probably say what I've tried:

Replaced fuel with fresh
Spark is happening
Plug looked ok, but I changed it anyway
Air filter reasonable
Oil recently changed (it's a 4-stroke B&S, from memory it says "Quantum" on top, but don't quote me on that)
Throttle on about halfway

What I get is:
1. Prime, pull rope, engine fires and runs for a few seconds, then dies as if starved for fuel.
2. Prime, pull rope, engine fires and runs for a few seconds, then dies as if starved for fuel.
3. Rinse, repeat.

I don't have Aerostart, but I do have have carby cleaner and tools. The carby is unknown territory for me, so I'll check youtube to see if anyone's done a vid of how to pull it apart when I get home.