Human-robot co-transportation allows for a human and a robot to perform an object transportation task cooperatively on a shared environment. This range of applications raises a great number of theoretical and practical challenges arising mainly from the unknown human-robot interaction model as well as from the difficulty of accurately model the robot dynamics. In this article, an adaptive impedance controller for human-robot co-transportation is put forward in task space. Vision and force sensing are employed to obtain the human hand position, and to measure the interaction force between the human and the robot. Using the latest developments in nonlinear control theory, we propose a robot end-effector controller to track the motion of the human partner under actuators' input constraints, unknown initial conditions, and unknown robot dynamics. The proposed adaptive impedance control algorithm offers a safe interaction between the human and the robot and achieves a smooth control behavior along the different phases of the co-transportation task. Simulations and experiments are conducted to illustrate the performance of the proposed techniques in a co-transportation task.