We prove that the Cauchy problem for the Benjamin-Ono-Burgers equation is uniformly globally well-posed in Hs (s? 1) for all ??[0,1]. Moreover, we show that as ??0 the solution converges to that of Benjamin-Ono equation in C([0,T]:Hs) (s?1) for any T>0. Our results give an alternative proof for the global well-posedness of the BO equation in H1(R) without using gauge transform, which was first obtained by Tao (2004) , and also solve the problem addressed in Tao (2004)  about the inviscid limit behavior in H1.