Quantum Corrections to the Weizs\"acker--Williams Distribution Function as Small x

We compute the quantum corrections to the gluon distribution function in the background of a non--Abelian Weizs\"acker--Williams field. These corrections are valid to all orders in the effective coupling $\alpha_s \mu$, where $\mu^2$ denotes the average valence quark color charge squared per unit area. We find $\ln(1/x)$ logarithmic corrections to the classical gluon distribution function. The one loop corrections to the classical Weizs\"acker-Williams field do not contribute to these singular terms in the distribution function. Their effect is to cause the running of $\alpha_s$.