One of the few firm predictions of string theory is the existence of a massless scalar field coupled to gravity, the dilaton. In its presence, the value of the fundamental constants of the universe, such as the fine-structure constant, will vary with the time-dependent vacuum expectation value of this field, in direct violation of the Einstein Equivalence Principle. The runaway dilaton proposed by Damour, Piazza, and Veneziano provides a physically motivated cosmological scenario which reconciles the existence of a massless dilaton with observations, while still providing non-standard and testable predictions. Furthermore, the field can provide a natural candidate for dynamical dark energy. While this model has been previously constrained from local laboratory experiments and low-redshift observations, we provide here the first full self-consistent constraints, also including high redshift data, in particular from the cosmic microwave background. We consider various possible scenarios in which the field could act as quintessence. Despite the wider parameter space, we make use of recent observational progress to significantly improve constraints on the model, showing that order unity couplings (which would be natural in string theory) are ruled out.