  We establish some conditional uniqueness of weak solutions to the viscous
primitive equations, and as an application, we prove the global existence and
uniqueness of weak solutions, with the initial data taken as small $L^\infty$
perturbations of functions in the space $X=\left\{v\in
(L^6(\Omega))^2|\partial_zv\in (L^2(\Omega))^2\right\}$; in particular, the
initial data are allowed to be discontinuous. Our result generalizes in a
uniform way the result on the uniqueness of weak solutions with continuous
initial data and that of the so-called $z$-weak solutions.
