We present a determination of the $P$-wave $\pi \pi \to \pi \gamma \star$ transition amplitude from lattice quantum chromodynamics. Matrix elements of the vector current in a finite-volume are extracted from three-point correlation functions, and from these we determine the infinite-volume amplitude using a generalization of the Lellouch-L"uscher formalism. We determine the amplitude for a range of discrete values of the $\pi \pi$ energy and virtuality of the photon, and observe the expected dynamical enhancement due to the $\rho$ resonance. Describing the energy dependence of the amplitude, we are able to analytically continue into the complex energy plane and from the residue at the $\rho$ pole extract the $\rho \to \pi \gamma \star$ transition form factor. This calculation, at $m\pi \approx 400$ MeV, is the first to determine the form factor of an unstable hadron within a first principles approach to QCD.