A precision-machined cross-slot flow geometry with a shape that has been optimized by numerical simulation of the fluid kinematics is fabricated and used to measure the extensional viscosity of a dilute polymer solution. Full-field birefringence microscopy is used to monitor the evolution and growth of macromolecular anisotropy along the stagnation point streamline, and we observe the formation of a strong and uniform birefringent strand when the dimensionless flow strength exceeds a critical Weissenberg number Wi_crit ≈ 0.5. Birefringence and bulk pressure drop measurements provide self-consistent estimates of the planar extensional viscosity of the fluid over a wide range of deformation rates (26 s⁻¹ ≤ ε ≤ 435 s⁻¹) and are also in close agreement with numerical simulations performed by using a finitely extensible nonlinear elastic dumbbell model.