A light-storage experiment with a total (storage and retrieval) efficiency η=56% is carried out by enclosing a sample, with a single-pass absorption of 10%, in an impedance-matched cavity. The experiment is carried out using the atomic frequency comb (AFC) technique in a praseodymium-doped crystal (0.05%Pr(3+):Y2SiO5) and the cavity is created by depositing reflection coatings directly onto the crystal surfaces. The AFC technique has previously by far demonstrated the highest multimode capacity of all quantum memory concepts tested experimentally. We claim that the present work shows that it is realistic to create efficient, on-demand, long storage time AFC memories.