Interfacing the different building blocks of a future large-scale quantum network will require efficient and noiseless frequency conversion of quantum light. Nitrogen-vacancy centers in diamond are a leading candidate to form the nodes of such a network. However, the performance of a suitable converter remains a bottleneck, with existing demonstrations severely limited by parasitic noise arising at the target telecommunication wavelength. Here we demonstrate a platform for efficient low-noise quantum frequency conversion based on a monolithic bulk periodically poled potassium titanyl phosphate cavity and show its suitability for the conversion of 637-nm single photons from nitrogen-vacancy centers in diamond to telecommunication wavelengths. By resonantly enhancing the power of an off-the-shelf pump laser, we achieve an internal conversion efficiency of $(72.3\ifmmode\pm\else\textpm\fi{}0.4)\mathrm{%}$ while generating noise of only $(110\ifmmode\pm\else\textpm\fi{}4)\phantom{\rule{0.2em}{0ex}}\mathrm{kHz}/\mathrm{nm}$ at the target wavelength without the need for any active stabilization. This constitutes a fivefold reduction in noise over existing state-of-the-art single-step converters at these wavelengths. We verify the almost-ideal preservation of nonclassical correlations by converting photons from a spontaneous-parametric-down-conversion source and moreover show the preservation of time-energy entanglement via Franson interferometry.