  We design and implement an atomic frequency comb quantum memory for 793 nm
wavelength photons using a monolithic cavity based on a thulium-doped
Y$_3$Al$_5$O$_{12}$ (Tm:YAG) crystal. Approximate impedance matching results in
the absorption of approximately $90\%$ of input photons and a memory efficiency
of (27.5$\pm$ 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 for the first time 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
large time-bandwidth product and multi-mode capacity for quantum networking.
