To overcome the drawbacks of Artificial Bee Colony ABC algorithm, which converges slowly in the process of searching and easily suffers from premature, this paper presents an effective approach, called ABC with accelerating convergence AC-ABC. In the process of evolution, first, the employed bee's position is regarded as the general centre position, the bees choose a location greedily as the new global optimal position in the original and general centre position; then we put the advantage of global optimal bee into evolution rule; we add the ability of best bee's learning into the standard ABC and reduce the value of convergence factor linearly according to the iteration times, which can improve the convergence of the new algorithm effectively. Experiments are conducted on 12 test functions to verify the performance of AC-ABC; the results demonstrate promising performance of our method AC-ABC on convergence velocity, precision, and stability of solution.