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.
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