  The present work investigates the Lesche stability (experimental robustness),
the thermodynamic stability, the Legendre structure of thermodynamics, and
derives the Maximum Entropy distribution of the one--parametric ``nonextensive
Gaussian'' entropy. We show that this entropy definition fulfills both
stability conditions for all values of its parameter ($q\in\mathbb{R}$). The
entropy maximizer contains the Lambert $W$--function, which allows the
preservation of the Legendre transformations.
