We use the diagrammatic Monte Carlo approach to solve the problem of a single spin-down fermion resonantly interacting with a Fermi gas of spin-up particles. Our solution is important for understanding the phase diagram and properties of the crossover from the BCS regime to the Bose-Einstein condensate in the strongly imbalanced regime. On the technical side, we develop a generic sign-problem-tolerant method for exact numerical solution of polaron-type models. This is a characteristic example of how Monte Carlo methods can be used to simulate divergent sign-alternating diagrammatic series.