This paper deals with the determination of the automorphism group of the metacyclic p-groups, P(p,m), given by the presentation P(p,m) = where p is an odd prime number. We show that Aut(P) has a unique Sylow p-subgroup, S_p, and that Aut(P) is isomorphic to the the semidirect product of S_p and Z_(p-1).