Let (Gt) be the random graph process (G0) is edgeless and Gt is obtained by adding a uniformly distributed new edge to Gt-1), and let tk denote the minimum time t such that the k-core of Gt (its unique maximal subgraph with minimum degree at least k) is nonempty. For any fixed k ≥3, the k-core is known to emerge via a discontinuous phase transition, where at time t = tk its size jumps from 0 to linear in the number of vertices with high probability (w.h.p.). It is believed that for any k ≥3, the core is Hamiltonian upon creation w.h.p., and Bollob́as, Cooper, Fenner and Frieze further conjectured that it in fact admits ≥(k - 1)/2 edgedisjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the k-core in G(n, p) was unknown for any k. We show here that for any fixed k ≥15, the k-core of Gt is w.h.p. Hamiltonian for all t ≥tk, that is, immediately as the k-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed k the k-core contains ≥(k - 3)/2 edge-disjoint Hamilton cycles w.h.p. for all t ≥tk.