This paper studies a continuous-time pursuit-evasion problem involving a single pursuer and a single evader on a plane. In contrast to other works that study this problem, we are interested in developing a pursuit strategy that doesn't require continuous, or even periodic, information about the position of the evader. To this end, we propose a self-triggered control strategy such that the pursuer can autonomously decide, based on out-dated information, when new samples of the evader's position is required in order to satisfy desired performance metrics. Our proposed algorithm guarantees capture of the evader in finite time with a finite number of sporadic updates without sacrificing any performance in terms of guaranteed time to capture as compared to classic algorithms that assume continuous information is available at all times. Simulations illustrate our results.