|In recent years, it has been shown that magnetic twist and axisymmetric MHD modes are ubiquitous in the solar atmosphere and therefore, the study of resonant absorption for these modes have become a pressing issue as it can have important consequences for heating magnetic flux tubes in the solar atmosphere and the observed damping. In this investigation, for the first time, we calculate the damping rate for axisymmetric MHD waves in weakly twisted magnetic flux tubes. Our aim is to investigate the impact of resonant damping of these modes for solar atmospheric conditions. This analytical study is based on an idealized configuration of a straight magnetic flux tube with a weak magnetic twist inside as well as outside the tube. By implementing the conservation laws derived by Sakurai et al. (1991) and the analytic solutions for weakly twisted flux tubes obtained recently by Giagkiozis et al. (2015), we derive the dispersion relation for the Alfv ́en continuum for axisymmetric modes. We also obtain an insightful analytical expression for the damping rate in the long wavelength limit. We consider an inhomogeneous layer where both the longitudinal magnetic field and the density are allowed to vary continuously, and we show that both have significant impact on the damping time. Given the conditions in the solar atmosphere, resonantly damped axisymmetric modes are highly likely to be ubiquitous and play an important role in energy dissipation. We also suggest that given the character of these waves, it is likely that they have already been observed in the guise of Alfvén waves.