An important research topic in survival analysis is related to the modeling and estimation of the cure rate, i.e. the proportion of subjects that will never experience the event of interest. However, most estimation methods proposed so far in the literature do not handle the case of insufficient follow-up, that is when the right end point of the support of the censoring time is strictly less than that of the survival time of the susceptible subjects, and consequently these estimators overestimate the cure rate in that case. We fill this gap by proposing a new estimator of the cure rate that makes use of extrapolation techniques from the area of extreme value theory. We establish the asymptotic normality of the proposed estimator, and show how the estimator works for small samples by means of a simulation study. We also illustrate its practical applicability through the analysis of data on the survival of breast cancer patients.