This paper presents a modified leader election algorithm for complete networks without sense of direction. The original algorithm, introduced by Villadangos et al. in [11], had the aim of reducing the number of exchanged messages in order to select a leader. However, the original O(n) algorithm fails to choose a leader on several occasions. A modified algorithm, which eliminates the problems that cause the wrong behaviour, is proposed. It is formally proved that the new algorithm verifies the correctness criteria that consist of selecting a unique leader in every case. Additionally, the algorithm maintains the O(n) complexity in both messages and time, where n is the number of nodes in the system.