Complexity of weighted cycle in a hamiltonian graph
If we are asking for a simple cycle the problem is NP-complete by a reduction from the Hamiltonian cycle problem. 

We want to find a Hamiltonian cycle in a graph $G$. We assign weight 1 to all edges of $G$, and add to this graph all the other edges with weight $\infty$. We have thus created a clique, which is obviously Hamiltonian, and we ask whether there exist a cycle with weight $n$ in it.