We demonstrate the first solid-state spin-wave optical quantum memory with on-demand read-out. Using the full atomic frequency comb scheme in a Pr3+∶Y2SiO5 crystal, we store weak coherent pulses at the single-photon level with a signal-to-noise ratio >10. Narrow-band spectral filtering based on spectral hole burning in a second Pr3+∶Y2SiO5 crystal is used to filter out the excess noise created by control pulses to reach an unconditional noise level of (2.0±0.3)×10−3 photons per pulse. We also report spin-wave storage of photonic time-bin qubits with conditional fidelities higher than achievable by a measure and prepare strategy, demonstrating that the spin-wave memory operates in the quantum regime. This makes our device the first demonstration of a quantum memory for time-bin qubits, with on-demand read-out of the stored quantum information. These results represent an important step for the use of solid-state quantum memories in scalable quantum networks.