  We prove a parametric h-principle for complete nonflat conformal minimal
immersions of an open Riemann surface $M$ into $\mathbb R^n$, $n\geq 3$. It
follows that the inclusion of the space of such immersions into the space of
all nonflat conformal minimal immersions is a weak homotopy equivalence. When
$M$ is of finite topological type, the inclusion is a genuine homotopy
equivalence. By a parametric h-principle due to Forstneric and Larusson, the
space of complete nonflat conformal minimal immersions therefore has the same
homotopy type as the space of continuous maps from $M$ to the punctured null
quadric. Analogous results hold for holomorphic null curves $M\to\mathbb C^n$
and for full immersions in place of nonflat ones.
