  We investigate the close relationship between minimal surfaces in Euclidean
3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as
in the case of minimal surfaces in Euclidean 3-space, the only complete
connected embedded constant mean curvature 1 surfaces with two ends in
hyperbolic space are well-understood surfaces of revolution -- the catenoid
cousins. In contrast to this, we show that, unlike the case of minimal surfaces
in Euclidean 3-space, there do exist complete connected immersed constant mean
curvature 1 surfaces with two ends in hyperbolic space that are not surfaces of
revolution -- the genus 1 catenoid cousins. The genus 1 catenoid cousins are of
interest because they show that, although minimal surfaces in Euclidean 3-space
and constant mean curvature 1 surfaces in hyperbolic 3-space are intimately
related, there are essential differences between these two sets of surfaces
(when the surfaces are considered globally). The proof we give of existence of
the genus 1 catenoid cousins is a mathematically rigorous verification that the
results of a computer experiment are sufficiently accurate to imply existence.
