From what you've described do the following:

1.  Have your battery charged fully then load tested.  You could have a dead
cell(s) and it's not holding a charge.
2.  Have your alternator tested.  If it's putting out anything less than
13.5volts while running it is bad and not charging your battery.

My bet is that your battery is most probably OK (but almost fully drained
right now) but  your alternator is bad.  However, there is a possibility
that both may be gone so don't be surprised if your mechanic or dealer says
to replace both.

Good luck - Jonathan

Exact same symptoms I had on my 2001 Sephia.  The alternator appeared to be
fine at idle, but as soon as it got over about 1500 RPM it would cut out.
That meant that the battery had to supply all of the electrical power
whenever it was driven.  The short term solution until I could get it to the
dealer was to put a battery charger on it every night.

When I finally took mine to the dealer they tried to tell me there was
nothing wrong with the charging system.  After a lengthy discussion with the
Service Manager I got them to test it again at highway speed RPM.  That
finally convinced them and they replaced the alternator.

Mike