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Binary black holes are the primary endpoint of massive stars. Their properties provide a unique opportunity to constrain binary evolution, which remains poorly understood. We predict the main properties of binary black holes and their merger products in/around the Milky Way. We present the first combination of a high-resolution cosmological simulation of a Milky Way-mass galaxy with a binary population synthesis model in this context. The hydrodynamic simulation, taken from the FIRE project, provides a cosmologically realistic star formation history for the galaxy, its stellar halo, and satellites. During post-processing, we apply a metallicity-dependent evolutionary model to the star particles to produce individual binary black holes. We find that 7 × 105 binary black holes have merged in the model Milky Way, and 1.2 × 106 binaries are still present, with a mean mass of 28M⊙. Because the black hole progenitors are strongly biased towards low-metallicity stars, half reside in the stellar halo and satellites and a third were formed outside the main galaxy. The numbers and mass distribution of the merged systems is broadly compatible with the LIGO/Virgo detections. Our simplified binary evolution models predict that LISA will detect more than 20 binary black holes, but that electromagnetic observations will be challenging. Our method will allow for constraints on the evolution of massive binaries based on comparisons between observations of compact objects and the predictions of varying binary evolution models. We provide online data of our star formation model and binary black hole distribution.