Mary in Kentucky said:

That line about the new moon rising would distract me so much I couldn't enjoy the image or the song. (To me, a new moon is one night each month as listed on my gardening calendar when there is no moon...a black circle on the calendar.)

From my unabridged The Random House Dictionary of the English Language, under "New moon":

the moon either when in conjunction with the Sun or soon after, being either invisible or visible only as a slender crescent.

Also, see Sir Patrick Spens:

"Late, late yestreen I saw the new moon,
Wi' the auld moon in her airms"

Dave Oesterreich