Removing singularities of minimal surfaces by isotopies

Given an open Riemann surface $M$, we show that the branch points and the complete ends of finite total curvature of a conformal minimal surface $M\to {\mathbb{R}}^n$, $n\ge 3$, can be removed by an isotopy through such surfaces. The analogous result holds for null holomorphic curves $M\to {\mathbb{C}}^n$.