Abstract : An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). In this paper, the authors prove that a'(Kp,p) = p+2 = Delta+2 when p is prime. They also show that if we remove any edge from Kp,p, the resulting graph is acyclically Delta + 1 = p + 1-edge-colorable.