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We establish anomalous inertial range scaling of structure functions for a model of advection of a passive scalar by a random velocity field. The velocity statistics is taken gaussian with decorrelation in time and velocity differences scaling as $|x|^{\kappa/2}$ in space, with $0\leq\kappa < 2$. The scalar is driven by a gaussian forcing acting on spatial scale $L$ and decorrelated in time. The structure functions for the scalar are well defined as the diffusivity is taken to zero and acquire anomalous scaling behavior for large pumping scales $L$. The anomalous exponent is calculated explicitly for the $4^{\m\rm th}$ structure function and for small $\kappa$ and it differs from previous predictions. For all but the second structure functions the anomalous exponents are nonvanishing.