Kinesin is a two-headed motor protein that moves processively along microtubules in discrete steps of -8 nm, driven by the hydrolysis of ATP. Molecular details of the mechanochemistry of this motor remain obscure. How is chemical energy stored in ATP coupled to mechanical displacement? Does the number of ATP molecules required for movement increase when kinesin has to work against an opposing force? To shed light on these questions, a force clamp was constructed based on a feedback-driven optical tweezers capable of maintaining constant loads on single moving kinesin molecules. This novel instrument provides unprecedented resolution of molecular motion and permits mechanochemical studies under controlled external loads. Analysis of records of kinesin motion under variable ATP concentrations and loads revealed several new features. 1) Kinesin stepping is tightly coupled to ATP hydrolysis over a wide range of forces, requiring only a single hydrolysis per 8 nm step. 2) Kinesin stall forces depend upon the ATP concentration. 3) Increased loads reduce the maximum velocity as anticipated but also raise the apparent Michaelis-Menten constant. This last result indicates that the kinesin cycle contains at least one load-dependent transition affecting the rate at which ATP molecules bind and subsequently commit to hydrolysis. But, it is likely that at least one other load-dependent rate exists, affecting turnover number. Together, these findings necessitate revisions to current models and understanding of kinesin motion.