For any prime p, all constacyclic codes of length p^s over the ring ℛ = F_p^m + uF_p^m are considered. The units of the ring ℛ are of the forms γ and ⍺ + uβ, where ⍺, β, and γ are nonzero elements of F_p^m, which provides p^m (p^m -1) such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length p^s over the finite field F_p^m are obtained; they are used as a tool to establish the structure and Hamming distances of all (⍺ + uβ)-constacyclic codes of length p^s over ℛ. We then classify all cyclic codes of length p^s over ℛ and obtain the number of codewords in each of those cyclic codes. Finally, a one-to-one correspondence between cyclic and γ-constacyclic codes of length p^s over ℛ is constructed via ring isomorphism, which carries over the results regarding cyclic codes corresponding to γ-constacyclic codes of length p^s over ℛ.