The pitch to the batter bounces off the ground and hits the batter. Should the batter be awarded first base?

Answer: Yes, batter is awarded first base.

Under Rule 2.00 of Official Baseball Rules, the definition of a ball is [emphasis added by me]:

A BALL is a pitch which does not enter the strike zone in flight and is not struck at by the batter.
Rule 2.00 (Ball) Comment: If the pitch touches the ground and bounces through the strike zone it is a “ball.” If such a pitch touches the batter, he shall be awarded first base. If the batter swings at such a pitch after two strikes, the ball cannot be caught, for the purposes of Rule 6.05 (c) and 6.09(b). If the batter hits such a pitch, the ensuing action shall be the same as if he hit the ball in flight.