While bright terahertz sources are used to perform nonlinear experiments, they can be advantageous for high-precision linear measurements of opaque samples. By placing the sample away from the focus, nonlinearities can be suppressed, and sizeable amounts of transmitted radiation detected. Here, this approach is demonstrated for a 0.5 mm thick layer of liquid water in a static sample holder. Variations of the index of refraction as small as (7 ± 2) · 10-4 were detected at 0.58 THz for an aqueous salt solution containing ten millimoles of sodium chloride. To my knowledge, this precision is unprecedented in time-domain spectroscopy studies of diluted aqueous systems or other optically thick and opaque materials.