When a compact object is formed in a binary, any mass lost during core collapse will impart a kick on the binary’s center of mass. Asymmetries in this mass loss or neutrino emission would impart an additional natal kick on the remnant black hole or neutron star, whether it was formed in a binary or in isolation. While it is well established that neutron stars receive natal kicks upon formation, it is unclear whether black holes do as well. Here, we consider the low-mass X-ray binary MAXI J1305-704, which has been reported to have a space velocity ≳200 km s ⁻¹ . In addition to integrating its trajectory to infer its velocity upon formation of its black hole, we account for recent estimates of its period, black hole mass, mass ratio, and donor effective temperature from photometric and spectroscopic observations. We find that if MAXI J1305-704 formed via isolated binary evolution in the thick Galactic disk, then the supernova that formed its black hole imparted a natal kick of at least 70 km s ⁻¹ while ejecting less than ≃1 M _⊙ with 95% confidence assuming uninformative priors on mass loss and natal kick velocity.