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 $delta f$ = 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 50 ns storage time we measure $eta_{textrm{e2e}}^{textrm{50ns}} = 3.4(3)%$ emph{end-to-end efficiency} of the fiber-coupled memory, with an emph{total intrinsic efficiency} $eta_{textrm{int}} = 17(3)%$. Straightforward technological improvements can boost the end-to-end-efficiency to $eta_{textrm{e2e}} approx 35%$; 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 readout noise level of $9cdot 10^{-3}$ photons is dominated by atomic fluorescence, and for input pulses containing on average $mu_{1}=0.27(4)$ photons the signal to noise level would be unity.
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