One of the most fundamental facts about the Fourier transform is the Hausdorff-Young inequality, which states that for any locally compact Abelian group, the Fourier transform maps $L^p$ boundedly to $L^q$, where the two exponents are conjugate and $p \in [1,2]$. For Euclidean space, the optimal constant in this inequality was found by Babenko for $q$ an even integer, and by Beckner for general exponents. Lieb showed that all extremizers are Gaussian functions. This is a uniqueness theorem; these Gaussians form the orbit of a single function under the group of symmetries of the inequality. We establish a stabler form of uniqueness for $1