  Writing $<R^2_N > = AN^{2\nu}(1+BN^{-\Delta_1}+CN^{-1}+ ...)$ for the mean
square end--to--end length $<R^2_N>$ of a self--avoiding polymer chain of $N$
links, we have calculated $\Delta_1$ for the two--dimensional {\em continuum}
case from a new {\em finite} perturbation method based on the ground state of
Edwards self consistent solution which predicts the (exact) $\nu=3/4$ exponent.
This calculation yields $\Delta_1=1/2$. A finite size scaling analysis of data
generated for the continuum using a biased sampling Monte Carlo algorithm
supports this value, as does a re--analysis of exact data for two--dimensional
lattices.
