A system for the communication of communication units (18, 38) between different communication networks (10, 30), comprising a) at least one central unit, via which communication units of a first communication network communicate with each other, b) another central unit, via which communication units of a second communication network communicate with each other, c) a computer-controlled gateway for coupling the central units (24, 46) of the communication networks (10, 30) to each other for the communication exchange between the communication networks (10, 30), d) wherein the computer-controlled gateway comprises - a storage unit for storing an address of a communication unit temporarily, which address has been called by a communication partner and which communication unit is assigned to another communication partner, - a conversion unit which adapts the address of the called communication partner, which address has been stored in the storage unit, to the respective communication network of the called communication partner, - a transmission unit which transmits the converted address to the central unit of the respective communication network which is assigned to the called communication partner for establishing a connection, - a register memory in which an address register for finding addresses that belong to different communication networks (10, 30) is provided, and wherein the system is characterized in that the gateway comprises - a special storage which is configured as an address storage, in which addresses that have been selected from the address register by a communication partner (16, 26) and which belong to personal contacts of this communication partner (16, 26) can be stored. a) at least one central unit, via which communication units of a first communication network communicate with each other, b) another central unit, via which communication units of a second communication network communicate with each other, c) a computer-controlled gateway for coupling the central units (24, 46) of the communication networks (10, 30) to each other for the communication exchange between the communication networks (10, 30), d) wherein the computer-controlled gateway comprises - a storage unit for storing an address of a communication unit temporarily, which address has been called by a communication partner and which communication unit is assigned to another communication partner, - a conversion unit which adapts the address of the called communication partner, which address has been stored in the storage unit, to the respective communication network of the called communication partner, - a transmission unit which transmits the converted address to the central unit of the respective communication network which is assigned to the called communication partner for establishing a connection, - a register memory in which an address register for finding addresses that belong to different communication networks (10, 30) is provided, and wherein the system is characterized in that the gateway comprises - a special storage which is configured as an address storage, in which addresses that have been selected from the address register by a communication partner (16, 26) and which belong to personal contacts of this communication partner (16, 26) can be stored. - a storage unit for storing an address of a communication unit temporarily, which address has been called by a communication partner and which communication unit is assigned to another communication partner, - a conversion unit which adapts the address of the called communication partner, which address has been stored in the storage unit, to the respective communication network of the called communication partner, - a transmission unit which transmits the converted address to the central unit of the respective communication network which is assigned to the called communication partner for establishing a connection, - a register memory in which an address register for finding addresses that belong to different communication networks (10, 30) is provided, and wherein the system is characterized in that the gateway comprises - a special storage which is configured as an address storage, in which addresses that have been selected from the address register by a communication partner (16, 26) and which belong to personal contacts of this communication partner (16, 26) can be stored.