Let $\g$ (resp.\ $\g'$) be a Lie algebra of dimension $d\le 3$ (resp.\ of finite dimension) over a field $k$ of characteristic $\not= 2$. We prove that $\g$ is isomorphic to $\g'$ as Lie algebras over $k$ if and only if the enveloping algebra $U(\g)$ of $\g$ is isomorphic to $U(\g')$ as $k$-algebras.