For any α < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class CtCx α that have nonempty, compact support in time on R × T3 and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for α < 1/3 due to Eyink and Constantin, E, Titi, solves Onsager's conjecture that the exponent α = 1/3 marks the threshold for conservation of energy for weak solutions in the class Lt ∞Cx α. The previous best results were solutions in the class CtCx α for α < 1/5, due to Isett, and in the class Lt 1 Cx α for α < 1/3 due to Buckmaster, De Lellis, Székelyhidi, both based on the method of convex integration developed for the incompressible Euler equations by De Lellis, Székelyhidi. The present proof combines the method of convex integration and a new “Gluing Approximation” technique. The convex integration part of the proof relies on the “Mikado flows” introduced by Daneri, Székelyhidi and the framework of estimates developed in the author's previous work.