Quantum memories matched to single photon sources will form an important cornerstone of future quantum network technology. We demonstrate such a memory in warm Rb vapor with on-demand storage and retrieval, based on electromagnetically induced transparency. With an acceptance bandwidth of δf=0.66 GHz, the memory is suitable for single photons emitted by semiconductor quantum dots. In this regime, vapor cell memories offer an excellent compromise between storage efficiency, storage time, noise level, and experimental complexity, and atomic collisions have negligible influence on the optical coherences. Operation of the memory is demonstrated using attenuated laser pulses on the single photon level. For a 50 ns storage time, we measure η_{e2e}^{50 ns}=3.4(3)% end-to-end efficiency of the fiber-coupled memory, with a total intrinsic efficiency η_{int}=17(3)%. Straightforward technological improvements can boost the end-to-end-efficiency to η_{e2e}≈35%; beyond that, increasing the optical depth and exploiting the Zeeman substructure of the atoms will allow such a memory to approach near unity efficiency. In the present memory, the unconditional read-out noise level of 9×10^{-3} photons is dominated by atomic fluorescence, and for input pulses containing on average μ_{1}=0.27(4) photons, the signal to noise level would be unity.