  The zero-momentum ghost-dilaton is a non-primary BRST physical state present
in every bosonic closed string background. It is given by the action of the
BRST operator on another state $\x$, but remains nontrivial in the semirelative
BRST cohomology. When local coordinates arise from metrics we show that dilaton
and $\x$ insertions compute Riemannian curvature and geodesic curvature
respectively. A proper definition of a CFT deformation induced by the dilaton
requires surface integrals of the dilaton and line integrals of $\x$.
Surprisingly, the ghost number anomaly makes this a trivial deformation. While
dilatons cannot deform conformal theories, they actually deform conformal
string backgrounds, showing in a simple context that a string background is not
necessarily the same as a CFT. We generalize the earlier proof of quantum
background independence of string theory to show that a dilaton shift amounts
to a shift of the string coupling in the field-dependent part of the quantum
string action. Thus the ``dilaton theorem'', familiar for on-shell string
amplitudes, holds off-shell as a consequence of an exact symmetry of the string
action.
