Pulse Code Modulation (PCM) signal streams are converted into Asynchronous Transfer Mode (ATM) cells for switching and transmission across a telecommunications network. Each cell carries one PCM sample of up to 48 different voice connections, the voice connections being selected from the PCM data streams because they have a common destination. The cells are transmitted over ATM virtual paths, each path transmitting one cell every 125 .mu.s. The ATM signals are switched by units for switching cells (cell switching), while keeping their payload contents intact, and by units for switching the signals for individual voice connections between cells of an ATM signal (cell remapping). Advantageously, ATM transmission systems interface with PCM systems without adding appreciable delay and without requiring additional buffering. Advantageously, new voice paths can be established most of the time by using available slots in the cells of existing virtual paths. Advantageously, large telecommunication networks can be implemented using a small number of switching units.Advantageously, such a cell remap unit provides a key element to the implementation of an efficient large ATM voice network.