  We establish sharp stability estimates of logarithmic type in determining an
impedance obstacle in $\mathbb{R}^2$. The obstacle is of general polygonal
shape and the impedance parameter can be variable. We establish the stability
results by using a single far-field pattern, which constitutes a longstanding
problem in the inverse scattering theory. This is the first stability result in
the literature in determining an impedance obstacle by a single far-field
measurement. If the obstacle is of a generally polygonal shape, the stability
in determining the obstacle is established in terms of a modified Hausdorff
distance and is independent of the boundary impedance parameter. If the
obstacle is further known to be convex, the stability in simultaneously
determining the obstacle and the boundary impedance is established in terms of
the classical Hausdorff distance. There are several technical novelties and
development in the mathematical strategy developed for establishing the
aforementioned stability results. First, the stability analysis is conducted
around a corner point in a micro-local manner. Second, our stability estimates
establish explicit relationships among the geometric configurations of the
obstacle and the vanishing order of the wave field at the corner point. Third,
we develop novel error propagation techniques to tackle singularities of the
wave field at a corner as well as to tackle the impedance boundary condition.
