Bought a small tube of thermal paste - came with a small plastic "spreader" for smoothing it out.
Carefully cleaned the CPU top and heatsink base with isopropyl alcohol, squished a blob of thermal mush on the CPU, spread it out *very* neatly. Put everything back together.
Turned it on (24 hours from last attempt [so cold]).
6 minutes 4 seconds then locked up again.
Turned off and back on again. 54 seconds to lock up.
Turned off and back on again - ooooooooh, different....one long beep and three short beeps from the speaker.
Turned off and back on again. 24 seconds, locked up.
Put it away, returned to this machine to a) drink coffee, b)post this and c) play Second Life.
Sigh.