We consider a square-lattice Ising model with a smooth inhomogeneity in the bulk, where the couplings deviate from the bulk critical coupling by Al-1, l being the distance from the center of the inhomogeneity. Scaling exponents describing decay of correlations along the inhomogeneity have been determined exactly by conformal invariance and finite-size scaling. It is shown that the homogeneous model (A=0) with bulk Ising exponents separates the first-order transition regime (A>0) from the region of nonuniversal criticality (A