Extensive studies have been made on the electronic voting in the last twenty years, and many schemes have been proposed, in which both the security as well as the effectiveness have been improved. However, most available secure vote schemes mainly focused on the simple "one man, one vote" plurality protocol. In this paper, we address the security issues in Borda voting protocol, another important social decision protocol, in which a voter can rank the candidates by assigning them different points. In this paper, we propose a secure Borda voting scheme that guarantees full privacy protection of the voters as well as universal verifiability and weak robustness. Instead of assuming of existence of trusted or semi-trusted authorities as in other secure voting schemes, we employ the homomorphic ElGamal encryption in our scheme and distribute the private key among the all voters to achieve full privacy protection of voters. In such a way, all voters jointly compute the outcome of the election without revealing any further information of voters? individual preferences. An analysis of the protocol against the security requirements shows that the new protocol achieves full privacy protection, public verifiability, weak robustness, in addition to others addressed by other protocols reported in the literature.