Among the accurate indoor localisation systems that are using WiFi, Bluetooth, or infrared technologies, the ones that are based on the GSM rely on a stable external infrastructure that can be used even in an emergency. This paper presents an accurate GSM indoor localisation system that achieves a median error of 4.39 metres in horizontal coordinates and up to 64 percent accuracy in floor prediction (for 84 percent of cases the floor prediction is mistaken by not more than a single floor). The test and reference measurements were made inside a six-floor academic building, with an irregular shape, whose dimensions are around 50 metres by 70 metres. The localisation algorithm uses GSM signal readings from the 7 strongest cells available in the GSM standard (or fewer, if fewer than 7 are available). We estimate the location by a three-step method. Firstly, we propose a point localisation solution (i.e., localisation based on only one measurement). Then, by applying the central tendency filters and the Multilayer Perceptron, we build a localisation system that uses a sequence of estimations of current and past locations. We also discuss major accuracy factors such as the number of observed signals or the types of spaces in the building.