Fast Field-Cycling MRI (FFC-MRI) is an emerging MRI technique that allows the main magnetic field to vary, allowing probing T1 at various magnetic field strengths. This technique offers promising possibilities but requires long scan times to improve the signal-to-noise ratio. This paper presents an algorithm derived from the two-point method proposed by Edelstein that can estimate T1 using only one image per field, thereby shortening the scan time by a factor of nearly two, taking advantage of the fact that the equilibrium magnetisation is proportional to the magnetic field strength. Therefore the equilibrium magnetisation only needs measuring once, then T1 can be found from inversion recovery experiments using the Bloch equations. The precision and accuracy of the algorithm are estimated using both simulated and experimental data, by Monte-Carlo simulations and by comparison with standard techniques on a phantom. The results are acceptable but usage is limited to the case where variations of the main magnetic field are fast compared with T1 and where the dispersion curve is relatively linear. The speed-up of T1-dispersion measurements resulting from the new method is likely to make FFC-MRI more acceptable when it is applied in the clinic.