The shortest-superstring problem is to find a shortest possible string that contains every string in a given set as substrings. This problem has applications to data compression and DNA sequencing. Since the problem is NP-hard and MAX SNP-hard, approximation algorithms are of interest. We present a new algorithm which always finds a superstring that is at most 2.89 times as long as the shortest superstring. Our result improves the 3-approximation result of Blum et al.