The statistical mechanics of finite L×L Ising square lattices in a field ±h of random sign is investigated numerically by a modified recursive transfer matrix method for 6<~L<~16. Our results are consistent with the absence of a spontaneous magnetization for h≠0 even in the ground state. The singularities occurring at T=0 in the range 0<~h<~4J, J being the exchange constant, are discussed in terms of a cluster expansion. For nonzero T, less than the critical temperature of the pure two-dimensional Ising model, and h sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for h→0 and is calculated from the domain-wall density.