In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface M into a minimally convex domain D ⊂ R 3 can be approximated, uniformly on compacts inM = M \bM , by proper complete conformal minimal immersionsM → D (see Theorems 1.1, 1.7, and 1.9). We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in R 3 (see Theorem 1.16), and we characterize the minimal surface hull of a compact set K in R n for any n ≥ 3 by sequences of conformal minimal discs whose boundaries converge to K in the measure theoretic sense (see Corollary 5.6).