We subjected 106 new high-resolution spectra of the double-lined spectroscopic close binary varepsilon Lup, obtained in a time-span of 17 days from two different observatories, to a detailed study of orbital and intrinsic variations. We derived accurate values of the orbital parameters. We refined the sidereal orbital period to 4.55970d days and the eccentricity to e=0.277. By adding old radial velocities, we discovered the presence of apsidal motion with a period of the rotation of apses of about 430 years. Such a value agrees with theoretical expectations. Additional data is needed to confirm and refine this value. Our dataset did not allow us to derive the orbit of the third body, which is known to orbit the close system in 64 years. We present the secondary of varepsilon Lup as a new β Cephei variable, while the primary is a β Cephei suspect. A first detailed analysis of line-profile variations of both primary and secondary led to detection of one pulsation frequency near 10.36 c d-1 in the variability of the secondary, while no clear periodicity was found in the primary, although low-amplitude periodicities are still suspected. The limited accuracy and extent of our dataset did not allow any further analysis, such as mode-identification.