  We compute the electromagnetic mass differences of mesons containing a single
heavy quark in terms of measurable data using QCD-based arguments in
heavy-quark effective theory. We derive an unsubtracted dispersion relation
that shows that the mass differences are calculable in terms of the properties
of the lowest-lying physical intermediate states. We then consider the problem
in the large-$N$ limit, where $N$ is the number of QCD colors. In this limit,
we can write a kind of double-dispersion relation for the amplitude required to
determine the electromagnetic mass difference. We use this to derive analogs of
the Weinberg sum rules for heavy meson matrix elements valid to leading order
in $1/N$ and to $O(1/m_Q)$ in the heavy quark expansion. In order to obtain our
final result, we assume that the electromagnetic mass differences and sum rules
are dominated by the lowest-lying states in analogy with the situation for the
$\pi^+$--$\pi^0$ mass difference. Despite the fact that some of the matrix
elements appearing in our final result have not yet been accurately measured,
we can obtain useful numerical estimates: for example, we obtain $(M_{B^+} -
M_{B^0})^{EM} \simeq +1.8 \MeV$. We argue that our results are accurate to
about $30\%$.
