I'm looking at a nice half moon today, easily visible in the bright blue afternoon sky on a brightly sunlit day. You can see the half of the moon that's lit by the sun, but the dark half is invisible.

If you'd waited until a while after sunset you'd also have seen the "invisible" half. That's because there is, in most of the orientations of the Sun-Earth-Moon system, sufficient light reflected from the earth's surface to reach the moon, illuminate even the part of its surface shaded from the sun and be reflected back to the earth. When there is very little haze you can even see the moon when it appears as only a fingernail and thus near "new".

I suspect it is an incompletely remembered understanding of this type of visibility that is at the bottom of Little Hawk's (and others') misapprehension that inspired his original question. And, while we're at it, the "perfect" full moon is an eclipsed one (a full lunar eclipse) and the light reflected from the moon's surface in that moment is sunlight that has been refracted through the earth's atmosphere, which is why it is copper-coloured. A full solar eclipse, conversely, is the "perfect" new moon, but it's not a good idea to look directly at it.

Cheers, Rowan