  We propose a new method to probe a possible time evolution of the fine
structure constant $\alpha$ from X-ray and Sunyaev-Zeldovich measurements of
the gas mass fraction ($f_{gas}$) in galaxy clusters. Taking into account a
direct relation between variations of $\alpha$ and violations of the
distance-duality relation, we discuss constraints on $\alpha$ for a class of
dilaton runaway models. Although not yet competitive with bounds from high-$z$
quasar absorption systems, our constraints, considering a sample of 29
measurements of $f_{gas}$, in the redshift interval $0.14 < z < 0.89$, provide
an independent estimate of $\alpha$ variation at low and intermediate
redshifts. Furthermore, current and planned surveys will provide a larger
amount of data and thus allow to improve the limits on $\alpha$ variation
obtained in the present analysis.
