We present high-resolution, time-resolved optical spectroscopy of the black hole X-ray transient Swift J1753.5-0127. Our optical spectra do not show features that we can associate with the companion star. However we do observe broad, double-peaked emission lines, typical of an accretion disc. We show that the mass of the compact object is likely $>7.4pm1.2M_{odot}$, much higher than previous suggestions of a low-mass ($<5M_{odot}$) black hole.