We study a single polaron in the Su-Schrieffer-Heeger (SSH) model using four different techniques (three numerical and one analytical). Polarons show a smooth crossover from weak to strong coupling, as a function of the electron-phonon coupling strength λ, in all models where this coupling depends only on phonon momentum q. In the SSH model the coupling also depends on the electron momentum k; we find it has a sharp transition, at a critical coupling strength λc, between states with zero and nonzero momentum of the ground state. All other properties of the polaron are also singular at λ=λc. This result is representative of all polarons with coupling depending on k and q, and will have important experimental consequences (e.g., in angle-resolved photoemission spectroscopy and conductivity experiments).