A ground based (1-g) experiment is in progress that measures the turbidity of the density-matched, binary fluid mixture Methanol-Cyclohexane extremely close to its liquid-liquid critical point. By covering the range of reduced temperatures t = (T-Tc) / Tc from 10-8 to 10-2, the turbidity measurements will allow the critical exponent eta to be determined. The transmitted and reference light intensities can be measured to 0.1 percent, the fluid temperature can be controlled to 6.5 μK near room temperature, and preliminary turbidity data show a non-zero eta consistent with theoretical predictions. No experiment has precisely determined a value of the critical exponent η, yet its value is significant to theorists in critical phenomena. Relatively simple critical phenomena, as in the liquid-liquid system studied here, serve as model systems for more complex behavior near a critical point. This research was supported by NASA grants NAG3-1404 and UGS97-0308 and C.A.W. and S.M.Y.L. acknowledge summer support from NSF DMR-9619406.