The performance of a caesium fountain frequency reference for use in precision measurements of trapped antihydrogen in the ALPHA experiment at CERN is evaluated. A description of the fountain is provided together with a characterisation of systematic effects. The impact of the magnetic environment in the Antimatter Factory, where the fountain is installed, on the performance of the fountain is considered and shown to be insignificant. The systematic fractional frequency uncertainty of the fountain is 3.0x10^-16. The short-term frequency stability of the measured frequency from the ALPHA-HM1 maser is 1.5x10^-13 𝜏^-1/2, whereas the fountain itself shows a stability limit of 4.7x10^-14 𝜏^-1/2. We find a fractional frequency difference of (1.0 ± 2.2 (stat.) ± 6.5 (syst.))x10^-16  in a comparison with Terrestrial Time via a GNSS Common View satellite link between January 2023 and June 2024. The fountain enables a significant increase in frequency precision in antihydrogen spectroscopic measurements, and paves the way for improved limits on matter–antimatter comparisons.