Summary: The uniqueness of the best rank-one approximation of a tensor under the Frobenius norm is a basic and important studying subject in the tensor theory. By introducing the quasi-singular value of a tensor, we present a new sufficient condition under which the best rank-one approximation of a nonzero tensor \(\underline{X}\in\mathbb{R}^{d_{1}\times d_{2}\times d_{3}}\) is unique and obtain that the set consisting of all tensors satisfying the sufficient condition is an open and dense set in \(\mathbb{R}^{d_{1}\times d_{2}\times d_{3}}\). In addition, we present a necessary and sufficient condition under which the best rank-one approximation of the sum tensor \(\underline{\tilde X}=\underline{X} + {\underline E}\) under some assumption on the tensor \(\underline{\tilde X}\in\mathbb{R}^{d_{1}\times d_{2}\times d_{3}}\) is equal to the best rank-one approximation of \({\underline X}\). Meanwhile, a numerical algorithm is proposed for computing the quasi-singular value of a tensor. Finally, several testing examples are presented to illustrate the feasibility of the computational process and the correctness of the theoretical results obtained in the paper.