We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u_s=-1 (where u=e^'4K) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A_s'(1+u)(- gamma _s') with exponent gamma _s'=3/2. The critical amplitude A_s' is calculated. Other critical exponents are found to be alpha _s'= sigma _s=0 and beta _s= 1/4 , so that the scaling relation alpha _s'+2 beta _s+ gamma _s' is satisfied. However, using exact results for beta _s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta _s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.