In a process for monitoring an object with an alarm detection unit allocated to the object and with an alarm central office, an alarm can be transmitted immediately from the alarm detection unit to the alarm central office over a digital data network without the need for costly dedicated lines because when an alarm is detected the calling party number of the decentralized alarm detection unit is transmitted as alarm information over the D channel of the ISDN network and the central office uses this calling party number to generate an alarm display before a connection has been established over the B channel of the ISDN network, for example in the form of an audio-video connection, to check the cause of the alarm.