Consider a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges that "hide" the cycle. Is it pos- sible to unravel the structure, that is, to efficiently find a Hamiltonian cycle in G? We describe an O(n3 log n) steps algorithm A for this purpose, and prove that it succeeds almost surely. Part one of A properly covers the "trouble spots" of G by a collection of disjoint paths. (This is the hard part to analyze.) Part two of A extends this cover to a full cycle by the rotation-extension technique which is already classical for such problems.