  We give the first correction to the suspension viscosity due to fluid
elasticity for a dilute suspension of spheres in a viscoelastic medium. Our
perturbation theory is valid to $O(\phi\mathrm{Wi}^2)$ in the Weissenberg
number $\mathrm{Wi}=\dot\gamma \lambda$, where $\dot\gamma$ is the typical
magnitude of the suspension velocity gradient, and $\lambda$ is the relaxation
time of the viscoelastic fluid. For shear flow we find that the suspension
shear-thickens due to elastic stretching in strain hot spots near the particle,
despite the fact that the stress inside the particles decreases relative to the
Newtonian case. We thus argue that it is crucial to correctly model the
extensional rheology of the suspending medium to predict the shear rheology of
the suspension. For uniaxial extensional flow we correct existing results at
$O(\phi\mathrm{Wi})$, and find dramatic strain-rate thickening at
$O(\phi\mathrm{Wi}^2)$. We validate our theory with fully resolved numerical
simulations.
