The version most familiar to us doesn't seem to be the original, and it is not favoured as the original tune. The one we are accustomed to hearing starts with the note on the third degree of the scale, with an accidental note going one half step down, then going back up one half step. Then it goes up one whole tone and one half tone to the fifth degree, then it drops down one hole tone and ends on the sixth degree, a minor sixth below the last note.
The other melody I heard was sung by the Cardiff Festival Choir, under the name O Little Town, O Little Town of Bethlehem. It's melody starts out on the fifth degree of the scale, then goes up a perfect fourth, rises then goes up one whole tone or major second, goes back donw a hole tone, back up a whole tone, and it goes up another whole tone. Then it goes up a half tone, and it goes up one whole tone, basically one octave from the first note of the tune.