Force Sensing and Force Control are essential to many industrial applications. Typically, a 6-axis Force/Torque (F/T) sensor is mounted between the robot's wrist and the end-effector in order to measure the forces and torques exerted by the environment onto the robot (the external wrench). Although a typical 6-axis F/T sensor can provide highly accurate measurements, it is expensive and vulnerable to drift and external impacts. Existing methods aiming at estimating the external wrench using only the robot's internal signals are limited in scope: for example, wrench estimation accuracy was mostly validated in free-space motions and simple contacts as opposed to tasks like assembly that require high-precision force control. Here we present a Neural Network based method and argue that by devoting particular attention to the training data structure, it is possible to accurately estimate the external wrench in a wide range of scenarios based solely on internal signals. As an illustration, we demonstrate a pin insertion experiment with 100-micron clearance and a hand-guiding experiment, both performed without external F/T sensors or joint torque sensors. Our result opens the possibility of equipping the existing 2.7 million industrial robots with Force Sensing and Force Control capabilities without any additional hardware.