  Heterotic vacua of string theory are realised, at large radius, by a compact
threefold with vanishing first Chern class together with a choice of stable
holomorphic vector bundle. These form a wide class of potentially realistic
four-dimensional vacua of string theory. Despite all their phenomenological
promise, there is little understanding of the metric on the moduli space of
these. What is sought is the analogue of special geometry for these vacua. The
metric on the moduli space is important in phenomenology as it normalises
D-terms and Yukawa couplings. It is also of interest in mathematics, since it
generalises the metric, first found by Kobayashi, on the space of gauge field
connections, to a more general context. Here we construct this metric, correct
to first order in alpha', in two ways: first by postulating a metric that is
invariant under background gauge transformations of the gauge field, and also
by dimensionally reducing heterotic supergravity. These methods agree and the
resulting metric is Kahler, as is required by supersymmetry. Checking that the
metric is in fact Kahler is quite intricate and uses the anomaly cancellation
equation for the H-field, in an essential way. The Kahler potential
nevertheless takes a remarkably simple form: it is Kahler potential for special
geometry with the Kahler form replaced by the alpha'-corrected hermitian form.
