The ultimate boundedness of a chaotic system is very important for the study of the qualitative behavior of a chaotic system. At present, explicit ultimate bound sets can be analytically obtained only for some special chaotic systems, and few results are known for hyperchaotic ones. In this paper, we investigate the ultimate boundedness for the new hyperchaotic Lorenz system by mathematical proof, and derive estimation of globally exponentially attractive set and positively invariant set. Using the estimation of the ultimate boundedness, we design a simple feedback controller to globally exponentially synchronize two hyperchaotic Lorenz systems, and present the results of numerical simulation.