  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
\emph{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's coupling parameters, showing
that order unity couplings (which would be natural in string theory) are ruled
out.
