The resonant scattering solar spectrophotometer `Mark-I', designed and build at the University of Birmingham (UK) and located at the Observatorio del Teide (Spain), has been continuously in operation for the past 38 years. During this period of time, it has provided high-precision measurements of the radial velocity of the Sun as a star, which has enabled the study of the small velocity fluctuations produced by the solar oscillations and the characterization of their spectrum. So far, it has been one of the pioneer experiments in the field of helioseismology and contributed to the development of that area. Moreover, because of its high-sensitivity and long-term instrumental stability, it also provides an accurate determination (to within a few parts in 103) of the absolute daily velocity offset, which contains the so-called solar gravitational red-shift. In this paper, results of the analysis of the measurements of this parameter over the whole period 1976-2013 are presented. The result of this series of measurements is 600.4 ± 0.8 m s-1 with an amplitude variation of ±5 m s-1, which is in anticorrelation with the phase of the solar activity cycle. The 5 per cent difference found with respect to the value predicted by the equivalence principle is probably due to the asymmetry of the solar spectral line used.