The ability of shape memory alloys (SMA) to respond to an external stimulus by modifying their dimensions can be used to generate motion or force in electromechanical devices and micro-machines. It has been often demonstrated that SMA-based devices are serious alternatives to conventional micrometric actuators. We have previously demonstrated that, using a high-quality position sensor, such as a linear variable differential transformer (LVDT), to provide the position feedback, accuracies about 3 μm in position control can be obtained. In this work, we present an actuator prototype based in a SMA wire, conceived to be used in lightweight applications, where the bulky position sensor previously used is replaced with a lighter alternative. The most convenient one, and also the most challenging, is to use the wire’s own resistance as a measure of its position, that is, to implement a sensorless control strategy. We propose to use a neural network to characterize the relation between the resistance of the wire and its strain and introduce this correspondence as the position feedback in a simple PID closed loop. The experimental results show that, in this way, accuracies about 70 μm can be obtained. The great advantage of this procedure is that the actuator is reduced to a single SMA element without any additional sensor, which is of great importance when the main goals are to reduce the overall weight, size, and cost of the actuator.