The deuteron form factor is calculated in a one-pion-exchange (OPE) approximation using single-variable unsubtracted dispersion relations in the squared-momentum-transfer variable. As a first step, the imaginary parts of the form factors are presented in terms of the four invariants of the deuteron-nucleon ($d-N$) vertex. The resulting equations are compared in detail with similar expressions obtained from the Jankus potential theory, and a clear understanding of the precise way in which the $d-N$ invariants play the role of the deuteron wave function emerges, enabling us to obtain relativistic wave functions. This comparison also shows the presence of new relativistic terms not included in the Jankus results (in particular, a new term appears in the magnetic moment). Then, the imaginary parts of the $d-N$ vertex invariants are calculated numerically. In this calculation the OPE contribution in its anomalous region is calculated exactly, while the contributions above the normal threshold are obtained by an educated guess based on sum rules which the invariants are assumed to satisfy. Finally, using these results, the form factors are calculated numerically. With the assumption of unsubtracted dispersion relations, it is necessary to assume only the charge and mass of the deuteron, and the pion-nucleon coupling constant, in order to completely determine the form factors. The numerical results for the magnetic and quadrupole moments agree with experiment to within 2%; the $D$-state probability is 5.5% but is not very reliably determined, and the other results, while less good, are still quite reasonable.

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