Let $M$ be a real-analytic manifold and let $T_{1,0}M\subset \mathbb C TM$ be a real-analytic abstract CR structure of hypersurface type. Equivalently, for some integer $m\ge 0$, $\dim_{\mathbb R}M=2m+1$, $T_{1,0}M$ is a real-analytic complex subbundle of rank $m$, $T_{0,1}M:=\overline{T_{1,0}M}$ satisfies $T_{1,0}M\cap T_{0,1}M=\{0\}$, and the sheaf of real-analytic local sections of $T_{0,1}M$ is closed under the Lie bracket. Then for every $p\in M$, there exist an open neighbourhood $U\subset M$ of $p$, an integer $N\ge 1$, and a real-analytic embedding $F:U\to \mathbb C^N$ whose component functions are CR functions on $U$.