We design and implement an atomic frequency comb quantum memory for 793-nm wavelength photons using a monolithic cavity based on a thulium- (Tm-) doped Y3Al5O12 crystal. Approximate impedance matching results in the absorption of 90% of input photons and a memory efficiency of (27.5 ± 2.7)% over a 500-MHz bandwidth. The cavity enhancement leads to a significant improvement over the previous efficiency in Tm-doped crystals using a quantum memory protocol. In turn, this allows us to store and recall quantum states of light in such a memory. Our results demonstrate progress toward efficient and faithful storage of single-photon qubits with a large time-bandwidth product and multimode capacity for quantum networking.