The effect of the wall-normal diffusion on the spanwise spreading of a steady passive scalar interface is computed for a laminar channel in which the Péclét number, $\Pe$, is high but the velocity profile is parabolic. Two regimes are found according to whether the dimensionless streamwise coordinate $\tilde{x}$ is smaller or larger than $\Pe$. In both cases the mixing layer spreads as $\tilde{x}^{1/2}$ to the lowest approximation in $\Pe^{-1}$, although with different numerical coefficients. When $\tilde{x},{\ll}\,\Pe$ there is a faster growth of order $\tilde{x}^{1/3}$ that is restricted to boundary layers near the wall. The intermediate region between those two limits is universal, and is computed numerically. Quantitative results are given that should be useful to experimentally measure diffusion coefficients. The results are easily generalizable to other velocity profiles.