In a cable-driven parallel robot (CDPR), force sensors are utilized at each winch motor to measure the cable tension in order to obtain the force distribution at the robot end-effector. However, because of the effects of friction in the pulleys and the unmodeled cable properties of the robot, the measured cable tensions are often inaccurate, which causes force-control difficulties. To overcome this issue, this paper presents an artificial neural network (ANN)-based indirect end-effector force-estimation method, and its application to CDPR force control. The pulley friction and other unmodeled effects are considered as black-box uncertainties, and the tension at the end-effector is estimated by compensating for these uncertainties using an ANN that is developed using the training datasets from CDPR experiments. The estimated cable tensions at the end-effector are used to design a P-controller to track the desired force. The performance of the proposed ANN model is verified through comparisons with the forces measured directly at the end-effector. Furthermore, cable force control is implemented based on the compensated tensions to evaluate the performance of the CDPR in wrench space. The experimental results show that the proposed friction-compensation method is suitable for application in CDPRs to control the cable force.