  It is well known that a magnetic monopole-electric charge system carries an
angular momentum in its electromagnetic fields. Here we show that in the Dirac
string formulation of magnetic charge the monopole-electric charge system also
carries a momentum in its electromagnetic fields. This overlooked field
momentum arises from the Coulomb electric field of the electric charge and the
solenoidal magnetic field of the Dirac string. This implies that the
monopole-charge system must either: (i) carry a ``hidden momentum" in the
string, indicating that the string is real, or (ii) that the monopole-charge
system violates the center-of-energy theorem. The overall conclusion is that
the Dirac monopole with a {\it regulated} Dirac string is not a true monopole,
and is not even a good effective description for topological monopoles ({\it
e.g.} the 't Hooft-Polyakov monopole outside the core).
