Quantum sensors based on the nitrogen-vacancy (NV) center in diamond are leading platforms for high-sensitivity magnetometry with nanometer-scale resolution. State-of-the-art implementations, however, typically rely on bulky free-space optics or sacrifice spatial resolution to achieve high sensitivities. Here, we realize an integrated platform that overcomes this trade-off by fabricating monolithic whispering-gallery-mode cavities from a diamond chip containing a high density of NV centers and by evanescently coupling excitation to and photoluminescence from the cavity using a tapered optical fiber. Employing a lock-in-amplified Ramsey magnetometry scheme, we achieve a photon-shot-noise-limited DC sensitivity of 58 nT Hz${}^{-1/2}$—the best sensitivity reported to date for a nanofabricated cavity-based magnetometer. The microscopic cavity size enables micrometer-scale spatial resolution and low-power operation, while fiber-coupling provides a path to scalable on-chip integration. Arrays of such sensors could enable NV NMR spectroscopy of sub-nanoliter samples, new magnetic-gradient imaging architectures, and compact biosensing platforms.