We constructed a sensitive ellipsometer to study extremely small ellipticities that are acquired by laser light when it traverses a multipass optical cavity in a 9-m-long superconducting magnet. This setup was used to measure the Cotton-Mouton constant (C(CM)) of neon at 514.5 nm. We find for 760 Torr (1 atm) and 25-degrees-C that C(CM)(Ne) = (5.5 +/- 0.3) x 10(-20) G-2 cm-1.