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Nitrogen-vacancy (NV) defect centers in diamond are promising solid-state magnetometers. Single centers allow for high-spatial-resolution field imaging but are limited in their magnetic field sensitivity. Using defect-center ensembles, sensitivity can be scaled with sqrt[N] when N is the number of defects. In the present work, we use an ensemble of N∼10^{11} defect centers within an effective sensor volume of 8.5×10^{-4}  mm^{3} for sensing at room temperature. By carefully eliminating noise sources and using high-quality diamonds with large NV concentrations, we demonstrate, for such sensors, a sensitivity scaling as 1/sqrt[t], where t is the total measurement time. The associated photon-shot-noise-limited magnetic-field sensitivity for ac signals of f=20  kHz is 0.9  pT/sqrt[Hz]. For a total measurement time of 100 s, we reach a standard deviation of about 100 fT. Further improvements using decoupling sequences and material optimization could lead to fT/sqrt[Hz] sensitivity.