We present a new measurement of the mass of the Milky Way (MW) based on observed properties of its largest satellite galaxies, the Magellanic Clouds (MCs), and an assumed prior of a LambdaCDM universe. A large, high resolution cosmological simulation of this universe provides a means to statistically sample the dynamical properties of bright satellite galaxies in a large population of dark matter halos. The observed properties of the MCs, including their circular velocity, distance from the center of the MW, and velocity within the MW halo, are used to evaluate the likelihood that a given halo would have each or all of these properties; the posterior PDF for any property of the MW system can thus be constructed. This method provides a constraint on the MW virial mass, 1.2(+0.7-0.4)e12 Msun (68% confidence), which is consistent with recent determinations that involve very different assumptions. In addition, we calculate the posterior PDF for the density profile of the MW and its satellite accretion history. Although typical satellites of 1e12 Msun halos are accreted over a wide range of epochs over the last 10 Gyr, we find a 72% probability that the Magellanic Clouds were accreted within the last Gyr, and a 50% probability that they were accreted together. Show Partial Abstract