Are the categories $U(\mathfrak{g})$-$\mathrm{mod}$ and $\mathfrak{g}$-$\mathrm{rep}$ equivalent?

Question

Let $\mathfrak{g}$ be a Lie algebra. Denote by $U(\mathfrak{g})$ its universal enveloping algebra.
Let $U(\mathfrak{g})$-$\mathrm{mod}$ be the category of modules over $U(\mathfrak{g})$, and $\mathfrak{g}$-$\mathrm{rep}$ be the category of $\mathfrak{g}$-representations.

Are the categories $U(\mathfrak{g})$-$\mathrm{mod}$ and $\mathfrak{g}$-$\mathrm{rep}$ equivalent?