Sorry, was not clear.

It is a Pentium 4 and will drain the battery in 1.5 hours or so, depending on
what's running on it.

When Ubuntu 10.04 is running, connecting or disconnecting the power supply will
cause it to sleep or hibernate, I'm not sure which.
Connecting or disconnecting power supply during startup, when 10.04 not yet
running, has no effect.

In whatever state it's in, the machine must be turned off and restarted to
revive it.

Hard to check with previous battery, which has truly failed. Changed battery and
upgraded about the same time.