  We propose a new discretization of a mixed stress formulation of the Stokes
equations. The velocity $u$ is approximated with
$H(\operatorname{div})$-conforming finite elements providing exact mass
conservation. While many standard methods use $H^1$-conforming spaces for the
discrete velocity, $H(\operatorname{div})$-conformity fits the considered
variational formulation in this work. A new stress-like variable $\sigma$
equalling the gradient of the velocity is set within a new function space
$H(\operatorname{curl} \operatorname{div})$. New matrix-valued finite elements
having continuous "normal-tangential" components are constructed to approximate
functions in $H(\operatorname{curl} \operatorname{div})$. An error analysis
concludes with optimal rates of convergence for errors in $u$ (measured in a
discrete $H^1$-norm), errors in $\sigma$ (measured in $L^2$) and the pressure
$p$ (also measured in $L^2$). The exact mass conservation property is directly
related to another structure-preservation property called pressure robustness,
as shown by pressure-independent velocity error estimates. The computational
cost measured in terms of interface degrees of freedom is comparable to old and
new Stokes discretizations.
