  We investigate the initial value problem for a defocusing nonlinear
Schr\"odinger equation with weighted exponential nonlinearity $$ i\partial_t
u+\Delta u=\frac{u}{|x|^b}(e^{\alpha|u|^2}-1); \qquad (t,x) \in
\mathbb{R}\times\mathbb{R}^2, $$ where $0< b <1$ and $\alpha=2\pi(2-b)$. We
establish local and global well-posedness in the subcritical and critical
regimes.
