In this paper a nonpolynomial Jacobi spectral-collocation (NJSC) method for the second kind Fredholm integral equations (FIEs) with weakly singular kernel $|s-t|^{-\gamma}$ is proposed. By dividing the integral interval symmetrically into two parts and applying the NJSC method symmetrically to the two weakly singular FIEs respectively, the mild singularities of the interval endpoints can be captured and the exponential convergence can be obtained. A detailed $L^{\infty}$ convergence analysis of the numerical solution is derived.  The NJSC method is then extended to two dimensional case and similar exponential convergence results are obtained for low regular solutions. Numerical examples are presented to demonstrate the efficiency of the proposed method.