DNA sequence comparisons can be used in a large number of applications in the field of bioinformatics. It is widely known that solving the string edit distance problem contributes to determining the degree of similarity between two sequences. Considering the importance of private data in DNA sequences, it is necessary to have a novel privacy-preserving protocol which will resolve the edit distance problem. In this paper, we present two different protocols guaranteeing accurate computation of the edit distance between two DNA sequences without disclosing their original DNA sequences. We introduce a Blind Third Party (BTP) which is an entity that allows communication between two parties without revealing private data by way of exchanging encrypted values. The BTP computes the edit distance without knowing Alice's and Bob's original sequence, and then it distributes the result to Alice and Bob. The two protocols that we developed have constant number of communications, which is much improvement compared to the edit distance protocol suggested by Atallah et al. whose number of communications increases proportional to the product of the length of two sequences.