A graphite-walled cavity ionization chamber of modular design and variable volume has been used to determine the air-kerma rate in the reference ⁶⁰Co field at the BIPM. The chamber can be configured in five sizes. High-accuracy mechanical measurements of the volume of the air cavity were made for each configuration using a co-ordinate measuring machine. Ionization current measurements were made for each configuration and corrected for the effects of ion recombination and diffusion, stem scatter and chamber orientation. Monte Carlo calculations of cavity dose were made to evaluate the correction factors k_wall and k_an. A reproducibility of the ionization current per mass of 1.5 parts in 10⁴ was achieved on the repeated assembly of each configuration. The results show an air-kerma rate determination that increases with volume, the total change being around 8 parts in 10⁴. When analysed differentially, the air-kerma rate relative to the BIPM standard is K_diff/K_BIPM = 1.0026(6). A detailed uncertainty budget is presented. Possible reasons for the observed behaviour are discussed that might have consequences for all existing standards for air-kerma.