Revealing the cosmic reionisation history is at the frontier of extragalactic astronomy. Cosmic microwave background (CMB) observations constrain the averaged epoch of reionisation. However, the detailed reionisation history, i.e., the ionisation fraction as a function of redshift, cannot be investigated by the CMB because the CMB observations are only sensitive to the amount of electrons integrated over redshift. Here we propose a method to directly measure the ionisation fraction of the intergalactic medium (IGM) as a function of redshift using fast radio bursts (FRBs). FRBs are new astronomical transients with millisecond timescales. Their dispersion measure (DM_IGM) is an indicator of the amount of ionised materials in the IGM. Since the differential of DM_IGM against redshift is proportional to the ionisation fraction, our method allows us to directly measure the reionisation history without any assumption on its functional shape. As a proof of concept, we constructed mock non-repeating FRB sources to be detected with the Square Kilometre Array, assuming three different reionisation histories with the same optical depth of Thomson scattering. We considered three cases of redshift measurements: (A) spectroscopic redshift for all mock data, (B) spectroscopic redshift for 10% of mock data, and (C) redshift estimated from an empirical relation of FRBs between their time-integrated luminosity and rest-frame intrinsic duration. In all cases, the reionisation histories are consistently reconstructed from the mock FRB data using our method, whereas the CMB method cannot distinguish between them. Our results demonstrate the capability of future FRBs in constraining the reionisation history.