The black hole (BH) candidate SWIFT J1753.5− 0127 has remained active since the onset of its 2005 outburst. Emission lines in the optical spectrumwere observed at the very beginning of the outburst, but since then the spectrum has been featureless making a precise BH mass estimation impossible. Here we present results from our optical and ultraviolet (UV) observations of SWIFT J1753.5− 0127 taken in 2012–2013. Our new observations show extremely broad, double-peaked emission lines in the optical and UV spectra. The optical data also show narrow absorption and emission features with nearly synchronous and significant Doppler motions. A radial velocity study of these lines which we associate with the secondary star, yields a semi-amplitude of K2 =  382 km s−1 . A time series analysis of the spectral and photometric data revealed a possible orbital periodicity of 2.85 h, significantly shorter than the reported 3.2 h periodic signal by Zurita et al. The observed variability properties argue against a low orbital inclination angle and we present several observational arguments in favour of the BH interpretation. However, the measured radial velocity semi-amplitude of the donor star and the short orbital period imply that SWIFT J1753.5− 0127 has one of the lowest measured mass function for a BH in a low-mass X-ray binary. We show that the compact object mass in excess of 5 Msun  is highly improbable. Thus, SWIFT J1753.5− 0127 is a BH binary that has one of the shortest orbital period and hosts probably one of the smallest stellar mass BH found to date.