FeSe is the focus of intense research interest because of its unusual non-magnetic nematic state and because it forms the basis for achieving the highest critical temperatures of any iron-based superconductor. However, its Cooper pairing mechanism has not been determined because an accurate knowledge of the momentum-space structure of superconducting energy gaps Ξiβ(k) on the different electron-bands Eiβ(k) does not exist. Here we use Bogoliubov quasiparticle interference (BQPI) imaging to determine the coherent Fermi surface geometry of the Ξ±- and Ξ΅-bands surrounding the Ξ=(0,0) and X=(Ο/aFeβ,0) points of FeSe, and to measure their superconducting energy gaps ΞΞ±β(k) and ΞΞ΅β(k). We show directly that both gaps are extremely anisotropic but nodeless, and are aligned along orthogonal crystal axes. Moreover, by implementing a novel technique we demonstrate the sign change between ΞΞ±β(k) and ΞΞ΅β(k). This complex configuration of ΞΞ±β(k) and ΞΞ΅β(k), which was unanticipated within pairing theories for FeSe, reveals a unique form of superconductivity based on orbital selective Cooper pairing of electrons from the dyzβ orbitals of iron atoms. This new paradigm of orbital selectivity may be pivotal to understanding the microscopic interplay of quantum paramagnetism, nematicity and high temperature superconductivity