The general quadratic group GQ_2n and its elementary subgroup EQ_2n are analogs in the theory of quadratic forms of the general linear group GL_n and its elementary subgroup E_n. This article proves that the stabilization map GQ_2n/EQ_2n → GQ_(2n+1)/EQ_2(n+1) is an isomorphism whenever n ≥ ΛS + 1 and ΛS denotes the Λ-stable rank of rings with anti-involution. As a corollary, a result is obtained which has been anticipated since the late 1960s: over rings of finite Bass-Serre dimension d, the stabilization map is an isomorphism whenever n ≥ d + 2.