We operate a nitrogen-vacancy (NV−) diamond magnetometer at ambient temperatures and study the dependence of its bandwidth on experimental parameters including optical and microwave excitation powers. A model based on the Bloch equations is used to analyze the NV center's response time, τ, during continuous optical and microwave irradiation, and τ−1 is shown to be a weighted average of T1−1 and T2−1, where T1 and T2 are the longitudinal and transverse relaxation times of the electron spin during optical irradiation. We measured a maximum detection bandwidth of ∼1.6 MHz with optical excitation intensity of ∼2.3 MW/cm2, limited by the available optical power. The sensitivity of the NV ensemble for continuous-wave magnetometry in the presence of photon shot noise is analyzed. Two detection schemes are compared, one involving modulation of the fluorescence by an oscillating magnetic field while the microwave frequency is held constant, and the other involving double modulation of the fluorescence when the microwave frequency is modulated during the detection. For the first of these methods, we measure a sensitivity of 4.6 ± 0.3 nT/√Hz, unprecedented in a detector with this active volume of ∼10 μm3 and close to the photon-shot-noise limit of our experiment. The measured bandwidth and sensitivity of our device should allow detection of micro-scale NMR signals with microfluidic devices.