We demonstrate the conditional detection of time-bin qubits after storage in and retrieval from a photon-echo-based waveguide quantum memory. Each qubit is encoded into one member of a photon pair produced via spontaneous parametric down-conversion, and the conditioning is achieved by the detection of the other member of the pair. By performing projection measurements with the stored and retrieved photons onto different bases, we obtain an average storage fidelity of 0.885±0.020, which exceeds the relevant classical bounds and shows the suitability of our integrated light-matter interface for future applications of quantum information processing.