We prove that for all k >= 4 and 1 <= l < k/2, every k-uniform hypergraph H on n vertices with delta(k-2)(H) >= (4(k-l)-1/4(k-l)(2) + o(1)) ((n)(2)) contains a Hamiltonian l-cycle if k-l divides n. This degree condition is asymptotically best possible. The case k = 3 was addressed earlier by Buss et al.