Thermal expansion can be a leading cause of uncertainty in length metrology. A cell-based refractometer has been designed at NIST which targets 10^-6} relative uncertainty in the measurement of helium refractivity; in terms of refractive index at ambient conditions, the accuracy goal is 3 x 10^-11} in refractive index. To achieve this level of accuracy, the length of the 0.5 m gas cell needs to be known within 100 nm. This is achievable when cell length is measured by coordinate-measuring machine at 20 ℃. However, the refractometer will operate at the thermodynamically known fixed-points of water and gallium, near 0 ℃ and 30 ℃, respectively. The cell is made from fused quartz glass, which has a nominal thermal expansion coefficient of 0.4 (µm/m)/K. Therefore, to scale the accuracy of the dimensional metrology across 20 ℃ to the triple-point of water requires that the thermal expansion coefficient of fused quartz glass is known within 10 (nm/m)/K, or 2.5 %. A method is described to measure the thermal expansion coefficient of fused quartz glass. The measurement principle is to monitor the change in resonance frequency of a Fabry--Perot cavity as its temperature changes; the Fabry--Perot cavity is made from fused quartz glass. The standard uncertainty in the measurement was less than 0.6 (nm/m)/K, or 0.15 %. The limit on performance is arguably uncertainty in the reflection phase-shift temperature dependence, because neither thermooptic nor thermal expansion coefficients of thin-film coatings are reliably known. However, several other uncertainty contributors are at the same level of magnitude, and so any improvement in performance would entail significant effort. Furthermore, measurements of three different samples revealed that material inhomogeneity leads to differences in the effective thermal expansion coefficient of fused quartz; inhomogeneity in thermal expansion among samples is 17 times larger than the measurement uncertainty in a single sample.