  In this short note we prove the logarithmic stability of the single
measurement uniqueness result for the fractional Calder\'on problem which had
been derived in \cite{GRSU18}. To this end, we use the quantitative uniqueness
results established in \cite{RS20a} and complement these bounds with a boundary
doubling estimate. The latter yields control of the order of vanishing of
solutions to the fractional Schr\"odinger equation. Then, following a scheme
introduced in \cite{S10,ASV13} in the context of the determination of a surface
impedance from far field measurements, this allows us to deduce logarithmic
stability of the potential $q$.
