  Previously we have established that the second Hamiltonian structure of the
KP hierarchy is a nonlinear deformation, called $\hat{W}_{\infty}$, of the
linear, centerless $W_{\infty}$ algebra. In this letter we present a free-field
realization for all generators of $\hat{W}_{\infty}$ in terms of two scalars as
well as an elegant generating function for the $\hat{W}_{\infty}$ currents in
the classical conformal $SL(2,R)/U(1)$ coset model. After quantization, a
quantum deformation of $\hat{W}_{\infty}$ appears as the hidden current algebra
in this model. The $\hat{W}_{\infty}$ current algebra results in an infinite
set of commuting conserved charges, which might give rise to $W$-hair for the
2d black hole arising in the corresponding string theory at level $k=9/4$.
