Interactions between atoms and lasers provide the potential for unprecedented control of quantum states. Fulfilling this potential requires detailed knowledge of frequency noise in optical oscillators with state-of-the-art stability. We demonstrate a technique that precisely measures the noise spectrum of an ultrastable laser using optical lattice-trapped $^{87}$Sr atoms as a quantum projection noise-limited reference. We determine the laser noise spectrum from near DC to 100 Hz via the measured fluctuations in atomic excitation, guided by a simple and robust theory model. The noise spectrum yields a 26(4) mHz linewidth at a central frequency of 429 THz, corresponding to an optical quality factor of $1.6times10^{16}$. This approach improves upon optical heterodyne beats between two similar laser systems by providing information unique to a single laser, and complements the traditionally used Allan deviation which evaluates laser performance at relatively long time scales. We use this technique to verify the reduction of resonant noise in our ultrastable laser via feedback from an optical heterodyne beat. Finally, we show that knowledge of our lasers spectrum allows us to accurately predict the laser-limited stability for optical atomic clocks.
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