  A new method for examining the possible space-time variation of the fine
structure constant ($\alpha$) is proposed. The technique uses a relatively
simple measurement with an optical resonator to compare atom-stabilized optical
frequency references. This method does not require that the exact frequency of
each reference be measured, and has the potential to yield more than a
1000-fold improvement in experimental sensitivity to changes in $\alpha$. A
specific realization of an experiment using this method is discussed which can
approach the precision of ${\dot\alpha/\alpha}\sim10^{-18}/\tau$, where $\tau$
is the measurement time. Moreover, for this specific realization, a measurement
of ${\dot\alpha/\alpha}\sim10^{-15}/{\rm yr}$ as a near-term goal is realistic.
