A numerical technique has been developed which enables the open-ended coaxial line sensor to be used for measurements on laminar dielectric specimens which are backed by a conducting plane. The theory, which allows the reflection coefficient of the sensor to be computed from the complex permittivity of the specimen, has been tested using an automatic network analyser for measurements on reference liquids at frequencies between 10 MHz and 3 GHz. The discrepancies between theory and measurement do not exceed the estimated measurement uncertainties, provided that the specimen is sufficiently lossy to render reflections from its perimeter negligible. The ability to measure laminar specimens is an important extension to the applications of the sensor in dielectric metrology.