The asymptotic stability of monotone decreasing kink profile solitary wave solutions of the compound KdV-Burgers equation was studied. The estimate of the first-order and second-order derivatives of monotone decreasing kink profile solitary wave solutions was obtained and the difficulties caused by nonlinear terms in the compound KdV-Burgers equation in the estimation were overcome by using the L² energy estimate method and Young's inequality. It is proved that the monotone decreasing kink profile solitary wave solution is asymptotically stable in H¹. Moreover, the decay rates of ψ in the sense of L² and L^∞ norm respectively are (1 + t)^-1/2 and (1 + t)^-1/4 by using the L² estimate method and Gargliado-Nirenberg inequality.