We report optical and infrared photometric and spectroscopic observations that identify the counterpart to the 358.6-s X-ray transient pulsar SAX J2103.5+4545 with a moderately reddened V=14.2 B0Ve star. This identification makes SAX J2103.5+4545 the Be/X-ray binary with the shortest orbital period known, Porb= 12.7 days. The amount of absorption to the system has been estimated to be AV=4.2±0.3, which for such an early-type star implies a distance of about 6.5 kpc. The optical spectra reveal major and rapid changes in the strength and shape of the Hα line. The Hα line was initially observed as a double peak profile with the ratio of the intensities of the blue over the red peak greater than one (V/R > 1). Two weeks later this ratio reversed (V/R< 1). Subsequently, in less than a month, the emission ceased and Hα appeared in absorption. This fast spectral variability is interpreted within the viscous decretion disc model and demonstrates the significant role of the neutron star on the evolution of the circumstellar disc around the Be star. The implications of the small orbit and moderate eccentricity on the spin period of the neutron star are discussed.