It oscillates nicely but does not have any power.
I know, it is not supposed to have much power but this one is weaker than
weak, the output couldn't drive any antenna.

I was wondering if there are AC-1 owners/builders around here who could
suggest a few things to try, who are knowlegable about this thing.

Did you do any voltage or current measurements? While I haven't played with a
real AC-1, I've built many similar 6V6 single tube transmitters. I have the
circuit for the AC-1, and it looks pretty similar to some I've built. First
thing to do is measure the B+ to the plate and screen. Next, open the plate
feed where it connects to the filter capacitor and insert a milliameter. With
the "load" capacitor at maximum, adjust the "tune" capacitor for minimum
current. Take a set of readings and get back to me.

Doug Moore KB9TMY (Formerly K6HWY)