In a companion paper by Koposov et al., RR Lyrae from textit{Gaia} Data Release 2 are used to demonstrate that stars in the Orphan stream have velocity vectors significantly misaligned with the stream track, suggesting that it has received a large gravitational perturbation from a satellite of the Milky Way. We argue that such a mismatch cannot arise due to any realistic static Milky Way potential and then explore the perturbative effects of the Large Magellanic Cloud (LMC). We find that the LMC can produce precisely the observed motion-track mismatch and we therefore use the Orphan stream to measure the mass of the Cloud. We simultaneously fit the Milky Way and LMC potentials and infer that a total LMC mass of $1.38^{+0.27}_{-0.24} times10^{11},rm{M_odot}$ is required to bend the Orphan Stream, showing for the first time that the LMC has a large and measurable effect on structures orbiting the Milky Way. This has far-reaching consequences for any technique which assumes that tracers are orbiting a static Milky Way. Furthermore, we measure the Milky Way mass within 50 kpc to be $3.80^{+0.14}_{-0.11}times10^{11} M_odot$. Finally, we use these results to predict that, due to the reflex motion of the Milky Way in response to the LMC, the outskirts of the Milky Ways stellar halo should exhibit a bulk, upwards motion.
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