Safety and dependability are of the utmost importance for physical human-robot interaction due to the potential risks that a relatively powerful robot poses to human beings. From the control standpoint, it is possible to improve safety by guaranteeing that the robot will never exhibit any unstable behavior. However, stability is not the only concern in the design of a controller for such a robot. During human-robot interaction, the resulting cooperative motion should be truly intuitive and should not restrict in any way the human performance. For this purpose, we have designed a new variable admittance control law that guarantees the stability of the robot during constrained motion and also provides a very intuitive human interaction. The former characteristic is provided by the design of a stability observer while the latter is based on a variable admittance control scheme that uses the time derivative of the contact force to assess human intentions. The stability observer is based on a previously published stability investigation of cooperative motion which implies the knowledge of the interaction stiffness. A method to accurately estimate this stiffness online using the data coming from the encoder and from a multiaxis force sensor at the end effector is also provided. The stability and intuitivity of the control law are verified in a user study involving a cooperative drawing task with a 3 degree-of-freedom (dof) parallel robot as well as in experiments performed with a prototype of an industrial Intelligent Assist Device.