  In this paper we prove that every bordered Riemann surface M admits a
complete proper null holomorphic embedding into a ball of the complex Euclidean
$3$-space $\mathbb{C}^3$. The real part of such an embedding is a complete
conformal minimal immersion $M\to \mathbb{R}^3$ with bounded image. For any
such $M$ we also construct proper null holomorphic embeddings $M\to
\mathbb{C}^3$ with a bounded coordinate function; these give rise to properly
embedded null curves $M\to SL_2(\mathbb{C})$ and to properly immersed Bryant
surfaces $M\to \mathbb{H}^3$ in the hyperbolic $3$-space. In particular, we
give the first examples of proper Bryant surfaces with finite topology and of
hyperbolic conformal type. The main novelty when compared to the existing
results in the literature is that we work with a fixed conformal structure on
$M$. This is accomplished by introducing a conceptually new method based on
complex analytic techniques. One of our main tools is an approximate solution
to the Riemann-Hilbert boundary value problem for null curves in
$\mathbb{C}^3$.
