We theoretically demonstrate that a pair of Dirac monopoles with opposite synthetic charges can be created within a single spin-1 Bose-Einstein condensate by steering the spin degrees of freedom by external magnetic fields. Although the net synthetic magnetic charge of this configuration vanishes, both the monopole and the antimonopole are accompanied by vortex filaments carrying opposite angular momenta. Such a Dirac dipole can be realized experimentally by imprinting a spin texture with a nonlinear magnetic field generated by a pair of coils in a modified Helmholtz configuration. We also investigate the case where the initial state for the dipole-creation procedure is pierced by a quantized vortex line with a winding number κ. It is shown that if κ=-1, the resulting monopole and antimonopole lie along the core of a singly quantized vortex whose sign is reversed at the locations of the monopoles. For κ=-2, the monopole and antimonopole are connected by a vortex line segment carrying two quanta of angular momentum, and hence the dipole as a whole is an isolated configuration. In addition, we simulate the long-time evolution of the dipoles in the magnetic field used to create them. For κ=0, each of the semi-infinite doubly quantized vortices splits into two singly quantized vortices, as in the case of a single Dirac monopole. For κ=-1 and κ=-2, the initial vortices deform into a vortex with a kink and a vortex ring, respectively.