We describe measurements of the motional sidebands produced by a mechanical oscillator (with effective mass 43 ng and resonant frequency 705 kHz) that is placed in an optical cavity and cooled close to its quantum ground state. The red and blue sidebands (corresponding to Stokes and anti-Stokes scattering) from a single laser beam are recorded simultaneously via a heterodyne measurement. The oscillator’s mean phonon number ¯n is inferred from the ratio of the sidebands, and reaches a minimum value of 0.84 ± 0.22 (corresponding to a mode temperature T = 28 ± 7μK). We also infer ¯n from the calibrated area of each of the two sidebands, and from the oscillator’s total damping. The values of ¯n inferred from these four methods are in close agreement. The behavior of the sidebands as a function of the oscillator’s temperature agrees well with theory that includes the quantum fluctuations of both the cavity field and the mechanical oscillator.