  In this paper, we prove that every confomal minimal immersion of an open
Riemann surface into $\mathbb{R}^n$ for $n\ge 5$ can be approximated uniformly
on compacts by conformal minimal embeddings. Furthermore, we show that every
open Riemann surface carries a proper conformal minimal embedding into
$\mathbb{R}^5$. One of our main tools is a Mergelyan approximation theorem for
conformal minimal immersions to $\mathbb{R}^n$ for any $n\ge 3$ which is also
proved in the paper.
