  We prove a vertical halfspace theorem for surfaces with constant mean
curvature $H={1/2},$ properly immersed in the product space $\h^2\times\re,$
where $\h^2$ is the hyperbolic plane and $\re$ is the set of real numbers. The
proof is a geometric application of the classical maximum principle for second
order elliptic PDE, using the family of non compact rotational $H=1/2$ surfaces
in $\h^2\times\re.$
