Abstract : We solve the Bj\"{o}rling problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve $\gamma$ and an analytic (timelike in the case of de Sitter three-space) unit vector field $N$ along and orthogonal to $\gamma$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature $1$ which contains $\gamma$ and the unit normal of which on $\gamma$ is $N$. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC $1$ surfaces.
Keywords : Bj\"{o}rling formula, constant mean curvature surfaces, de Sitter space
