The long-term behavior of longitudinal spin order is investigated analytically in infinite one-dimensional (1-D) chains of nuclear spins 1/2 driven by a multiple pulse sequence of nonresonant rf pulses in the presence of a strong magnetic field. We consider a chain of spins with only nearest-neighbor dipolar interactions, initially at thermal equilibrium in the rotating frame; and evolving under the Hamiltonian, H=Delta I-Z+H+2+H-2, (where Delta is a resonance offset of the carrying frequency of the pulses from the Larmor frequency of the spins, and H+2+H-2 is a two-spin/two-quantum nonsecular average dipolar Hamiltonian). The corresponding spin density operator and longitudinal magnetization are calculated exactly with fermion field operator representations of spin-1/2 operators. The spin density operator is determined by an additional integral of motion, beyond the energy, reflecting inherent nonergodic behavior. The establishment of the spin thermodynamic equilibrium between the Zeeman and the nonsecular dipolar reservoirs is examined. It is shown that such equilibrium is only possible in an external magnetic field H-0 much greater than H-loc (where H-loc is the local dipolar field). A discussion of ergodicity in 1-D spin chains with the full dipolar Hamiltonian, and a suggestion of possible NMR measurements of longitudinal magnetization to reveal broken ergodicity in the spin dynamics of materials with quasi-1-D distributions of spins are presented.