Battery dying, not charging
 

I have a 2001 1.8i petrol Accord that doesn't seem to be charging the battery. It died on me yesterday, instrument displays started to go off, then couldn't start, battery was down to 9V. I rang a relation and they bought a new battery which got it started and got me home, but only just as that too started to die.

I have measured the voltage on the battery and its 12.5 volts, even with the engine running. So I would assume the alternator has failed.

However I have my doubts, as the other morning on the way to work, I had the battery fail light come on, as well as abs, srs etc, looked as though battery was dying, then after about a minute it all came back to life, and was ok for 4 days.

If I disconnect the +ve lead from the alternator and started the engine, should it produce a voltage, or is it only when under load?

The alternator is not noisy, no obvious signs of failure.