We measure the variation of the escape speed of the Milky Way across a range of ∼40 kpc in Galactocentric radius. The local escape speed is found to be 521+46−30kms−1, in good agreement with other studies. We find that this has already fallen to 379+34−28kms−1 at a radius of 50 kpc. Through measuring the escape speed and its variation, we obtain constraints on the Galactic mass profile and rotation curve. The gradient in the escape speed suggests that the total mass contained within 50 kpc is 30+7−5×1010M⊙, implying a relatively light dark halo for the Milky Way. The local circular speed is found to be vc(R0)=223+40−34kms−1 and falls with radius as a power law with index −0.19 ± 0.05. Our method represents a novel way of estimating the mass of the Galaxy, and has very different systematics to more commonly used models of tracers, which are more sensitive to the central parts of the halo velocity distributions. Using our inference on the escape speed, we then investigate the orbits of high-speed Milky Way dwarf galaxies. For each considered dwarf, we predict small pericentre radii and large orbital eccentricities. This naturally explains the large observed ellipticities of two of the dwarfs, which are likely to have been heavily disrupted at pericentre.