Nuclear processes involving momenta much below the mass of the pion may be described by an effective field theory in which the pions do not appear as explicit degrees of freedom. The effects of the pion and all other virtual hadrons are reproduced by the coefficients of gauge-invariant local operators involving the nucleon field. Nucleon-nucleon scattering phase shift data constrains many of the coefficients that appear in the effective Lagrangean but at some order in the expansion coefficients enter that must be constrained by other observables. We compute several observables in the two-nucleon sector up to next-to-next-to leading order in the effective field theory without pions, or to the order at which a counterterm involving four-nucleon field operators is encountered. Effective range theory is recovered from the effective field theory up to the order where relativistic corrections enter or where four-nucleonexternal current local operators arise. For the deuteron magnetic moment, quadrupole moment and the np → dγ radiative capture cross section a fournucleon-one-photon counterterm exists at next-to-leading order. The electric polarizability and electric charge form factor of the deuteron are determined up to next-to-next-to-leading order, which includes the first appearance of relativistic corrections.