A Fermi gas of cold atoms allows precise control over the dimensionless effective range, $kFReff$, of the Feshbach resonance. Our pseudopotential formalism allows us to create smooth potentials with effective range, $-2\le kFReff\le 2$, which we use for a variational and diffusion Monte Carlo study of the ground state of a unitary Fermi gas. We report values for the universal constants of $\xi =0.388(1)$ and $\zeta =0.087(1)$, and compute the condensate fraction, momentum distribution, and pair correlations functions. Finally, we show that a gas with $kFReff1.9$ is thermodynamically unstable.