This paper proves the orbital asymptotic stability of solitary waves for the Amick-Schonbek Boussinesq system, a model for weakly nonlinear long surface waves in shallow water. For initial data sufficiently close to a solitary wave in the weighted Sobolev space H = (H^2 ∩ L^{2,1})^2, the global solution decomposes into a modulated solitary wave plus a remainder U(t) that decays locally to zero at rate t^{-1/2} log t in L^infinity. The modulation speed satisfies uniform bounds |c(t)-c_0| ≤ C ε_0 and |c'(t)| ≤ C ε_0^2 (1+t)^{-1}. Full convergence of c(t) to a limit is not proved; it remains open and is stated as a conditional corollary requiring |c'| in L^1 (e.g., if |c'(t)| = O((1+t)^{-1-ε}) for some ε>0). The work confirms the orbital part of Conjecture 1 of Klein-Saut (2024) and provides a complete spectral analysis of the linearized generator A_c = J L_c + c ∂_z, including an Evans function argument, a limiting absorption principle, and sharp t^{-1/2} dispersive estimates for the continuous projection Q_c.