  A virial theorem is established for the operator proposed by Brown and
Ravenhall as a model for relativistic one-electron atoms. As a consequence, it
is proved that the operator has no eigenvalues greater than $\max(m c^2, 2
\alpha Z - \frac{1}{2})$, where $\alpha$ is the fine structure constant, for
all values of the nuclear charge $Z$ below the critical value $Z_c$: in
particular there are no eigenvalues embedded in the essential spectrum when $Z
\leq 3/4 \alpha$. Implications for the operators in the partial wave
decomposition are also described.
