If the MB's BIOS automatically regulates CPU fan speed, it would never drop to zero rpm. Otherwise, either the CPU would burn up(hopefully that has not happened), or the BIOS would shut down the system if it has the option to do so when the CPU fan fails.
First, do not run the system if the CPU fan does not work at all. Even though the CPU is rated for around 90 C max, it may be irreversibly damaged if it gets above 70 C.
If the fan runs when the system is first started, and the fan stops and the system shuts down after a few seconds, then see if the MB's fan terminal has 3 pins. If so, then make sure the fan's connector also has 3 slots. If it only has 2 slots, then install a fan that has a 3-slot connector. The 3rd slot is for speed sensing, and if the fan's connector has only 2 slots, then the BIOS receives a zero fan speed signal, thinks that the fan has failed, and shuts down the system to protect the CPU from becoming overheated and damaged.
To temporarily work around this, reset CMOS either with a MB jumper(see MB's documentation), or by removing the CMOS battery for at least 30 minutes. This will disable the CPU fan failure option.
If that does not apply, then there should be another fan terminal on the MB. See if the fan works with that one. If it does, then the other fan terminal is probably bad.