Let F be a non-archimedean local field of characteristic 0 and ψ a nontrivial additive character. Weil first defined the Weil index γ(a, b) (a ∈ F*) in his famous paper, from which we know that γ(a,ψ)γ(b,ψ) ＝γ(ab,ψ)γ(1,ψ) (a, b) and γ(a,ψ)4 ＝ (- 1, - 1),where (a, b) is the Hilbert symbol for F. The Weil index plays an important role in the theory of theta series and in the general representation theory. In this paper, we establish an identity relating the Weil index γγ(a,ψ) and the Gauss sum.