The trajectory of a colloidal sphere diffusing in water records a history of the random forces exerted on the sphere by thermally-driven fluctuations in the suspending fluid. The trajectory therefore can be used to characterize the spectrum of thermal fluctuations, and thus to obtain an estimate for Boltzmann's constant. We demonstrate how to use holographic video microscopy to track a colloidal's sphere's three-dimensional motions with nanometer precision while simultaneously measuring its radius to within a few nanometers. The combination of tracking and characterization data reliably yields Boltzmann's constant to within two percent, and also provides the basis for many other useful and interesting measurements in statistical physics, physical chemistry, and materials science.