  The Euclidean $(\phi^{4})_{3,\epsilon$ model in $R^3$ corresponds to a
perturbation by a $\phi^4$ interaction of a Gaussian measure on scalar fields
with a covariance depending on a real parameter $\epsilon$ in the range $0\le
\epsilon \le 1$. For $\epsilon =1$ one recovers the covariance of a massless
scalar field in $R^3$. For $\epsilon =0$ $\phi^{4}$ is a marginal interaction.
For $0\le \epsilon < 1$ the covariance continues to be Osterwalder-Schrader and
pointwise positive. After introducing cutoffs we prove that for $\epsilon > 0$,
sufficiently small, there exists a non-gaussian fixed point (with one unstable
direction) of the Renormalization Group iterations. These iterations converge
to the fixed point on its stable (critical) manifold which is constructed.
