We investigate consequences of an ultraviolet fixed point in quantum gravity for the cosmological constant. For this purpose we perform dimensional reduction of a general dilatation-symmetric effective action $\ensuremath{\Gamma}$ in dimension $d>4$ to an effective four-dimensional theory of gravity with a dilaton field. We find a stable flat phase in the space of the extrema of $\ensuremath{\Gamma}$ which results in a vanishing four-dimensional cosmological constant $\ensuremath{\Lambda}$. In order to understand the self-tuning mechanism leading to $\ensuremath{\Lambda}=0$ we discuss in detail the most general warped geometries with maximal four-dimensional symmetry and $SO(d\ensuremath{-}4)$ isometry of internal space. While the solutions of the $d$-dimensional field equations admit singular spaces with arbitrary $\ensuremath{\Lambda}$, the extremum condition for $\ensuremath{\Gamma}$ imposes additional restrictions which result in $\ensuremath{\Lambda}=0$. In cosmology, the dilatation-symmetric fixed point may only be reached for asymptotic time $t\ensuremath{\rightarrow}\ensuremath{\infty}$. At finite $t$ dilatation anomalies result in an effective potential and mass for the pseudodilaton or cosmon and in dark energy.