  We formulate a renormalizable quantum gravity in $2+\epsilon$ dimensions by
generalizing the nonlinear sigma model approach to string theory. We find that
the theory possesses the ultraviolet stable fixed point if the central charge
of the matter sector is in the range $0~<~c~<~25$. This may imply the existence
of consistent quantum gravity theory in 3 and 4 dimensions. We compute the
scaling dimensions of the relevant operators in the theory at the ultraviolet
fixed point. We obtain a scaling relation between the cosmological constant and
the gravitational constant, which is crucial for searching for the continuum
limit in the constructive approach to quantum gravity.
