Context. HD 13724 is a nearby solar-type star at 43.48 ± 0.06 pc hosting a long-period low-mass brown dwarf detected with the CORALIE echelle spectrograph as part of the historical CORALIE radial-velocity search for extra-solar planets. The companion has a minimum mass of 26.77 −2.2 +4.4 M Jup and an expected semi-major axis of ~240 mas making it a suitable target for further characterisation with high-contrast imaging, in particular to measure its inclination, mass, and spectrum and thus establish its substellar nature. Aims. Using high-contrast imaging with the SPHERE instrument on the Very Large Telescope (VLT), we are able to directly image a brown dwarf companion to HD 13724 and obtain a low-resolution spectrum. Methods. We combine the radial-velocity measurements of CORALIE and HARPS taken over two decades and high-contrast imaging from SPHERE to obtain a dynamical mass estimate. From the SPHERE data we obtain a low-resolution spectrum of the companion from Y to J band, as well as photometric measurements from IRDIS in the J , H , and K bands. Results. Using high-contrast imaging with the SPHERE instrument at the VLT, we report the first images of a brown dwarf companion orbiting the host star HD 13724. It has an angular separation of 175.6 ± 4.5 mas and an H -band contrast of 10.61 ± 0.16 mag, and using the age estimate of the star to be ~1 Gyr gives an isochronal mass estimate of ~44 M Jup . By combining radial-velocity and imaging data we also obtain a dynamical mass of 50.5 −3.5 +3.3 M Jup . Through fitting an atmospheric model, we estimate a surface gravity of log g = 5.5 and an effective temperature of 1000 K. A comparison of its spectrum with observed T dwarfs estimates a spectral type of T4 or T4.5, with a T4 object providing the best fit.