  In this paper we show the Wilson effective action for the 2-dimensional
O(N+1)-symmetric lattice nonlinear sigma-model computed in the 1-loop
approximation for the nonlinear choice of blockspin $\Phi(x)$, $\Phi(x)=
\Cav\phi(x)/{|\Cav\phi(x)|}$,where $\Cav$ is averaging of the fundamental field
$\phi(z)$ over a square $x$ of side $\tilde a$.
  The result for $S_{eff}$ is composed of the classical perfect action with a
renormalized coupling constant $\beta_{eff}$, an augmented contribution from a
Jacobian, and further genuine 1-loop correction terms. Our result extends
Polyakov's calculation which had furnished those contributions to the effective
action which are of order $\ln \tilde a /a$, where $a$ is the lattice spacing
of the fundamental lattice. An analytic approximation for the background field
which enters the classical perfect action will be presented elsewhere.
