The "with" part of "with one normal melee attack" says to me this modifies an attack. Is there some reason in the RAW that a Charge or an attack as part of a full iterative attack would not be considered "normal".
For some weird reason, I did not consider that a Paladin could use more than one in a round, probably because I am in the Plane Sailing school of thought.
But a Hasted Paladin with Extra Smiting could be very, very frightening for 1 round. Yeeowch!