The Be star HR 1960 has been monitored from 1982 to 1998 in Geneva photometry and from November 1989 to March 1993 by the Hipparcos satellite. This star shows two kinds of luminosity variation: i) The mean luminosity in V has slightly increased, by 5 millimags, during the survey (long-term variation); ii) A periodic light variation with a period of 395 d has been observed, in both Geneva and Hipparcos photometries (mid-term variation). The periodic variation is characterized by extremely small semi-amplitudes, 3 millimags in V and 2 millimags in [B −V ]. This is most probably the long-period variable star with the smallest amplitude yet known. The V and [B − V ] curves are anti-correlated, i.e. the star is found to be redder when brighter in the Paschen continuum. Both types of variability are attributed to the Be characteristics of the star. A model similar to that of Carrier et al. (1999) for HR 2968 is proposed: the Be star could be the main component of a binary system having an orbit of period 395 d; it would be surrounded by matter expelled in its equatorial plane and the variability would results from the gravitational and/or radiative interaction of its companion star with this disk. The physical parameters of HR 1960 are determined by using Geneva photometric calibrations and the Hipparcos parallax.