Metadynamics is a versatile and capable enhanced sampling method for the computational study of soft matter materials and biomolecular systems. However, over a decade of application and several attempts to give this adaptive umbrella sampling method a firm theoretical grounding prove that a rigorous convergence analysis is elusive. This Letter describes such an analysis, demonstrating that well-tempered metadynamics converges to the final state it was designed to reach and, therefore, that the simple formulas currently used to interpret the final converged state of tempered metadynamics are correct and exact. The results do not rely on any assumption that the collective variable dynamics are effectively Brownian or any idealizations of the hill deposition function; instead, they suggest new, more permissive criteria for the method to be well behaved. The results apply to tempered metadynamics with or without adaptive Gaussians or boundary corrections and whether the bias is stored approximately on a grid or exactly.