The Pierre Auger Collaboration has developed a method for estimating the depth of shower maximum ($X_{\max}$) using the risetime of the signals recorded by the array of water-Cherenkov detectors. The average value of $X_{\max}$ is determined for energies above 3 EeV, using air showers with zenith angles up to 60$^{\circ}$. The work described here considerably extends that reported previously with an increase in the total number of events by a factor 2.1. In total 125,005 events are used, of which 237 are above 50 EeV. The data set is approximately a factor 30 larger than what is available from fluorescence measurements for energies above 3 EeV. The results reinforce previous conclusions: the mean mass of high-energy cosmic rays begins to rise above 3 EeV and extend that finding to higher energies with greater precision. Additionally, we find strong discrepancies when we make a comparison between observations and predictions from simulations with values which are obtained from measurements of the risetime of the recorded signals.