  A Green's-function theory of antiferromagnetic short-range and long-range
order (LRO) in the $S=1/2$ quasi-two-dimensional easy-plane XXZ model is
presented. As the main new result, {\it two} phase transitions due to the
combined influence of spatial and spin anisotropy are found, where below the
higher and lower N\'{e}el temperature there occurs LRO in the transverse and in
both the transverse and longitudinal spin correlators, respectively. Comparing
the theory with neutron-scattering data for the correlation length of $\rm
La_2CuO_4$, a very good agreement in the whole temperature dependence is
obtained. Moreover, for $\rm La_2CuO_4$, $\rm Sr_2CuO_2Cl_2$, and $\rm
Ca_{0.85}Sr_{0.15}CuO_2$ the second phase with longitudinal LRO is predicted to
appear far below room temperature.
