See if the mouse works by doing a clean boot:
If it does, then follow the steps in the above link to determine the cause of the conflict.
If it does not work after doing a clean boot, then its drivers may be missing or corrupted.
Start in Safe mode, open Device Manager, click the + next to 'Mouse', double click 'PS/2 Compatible Mouse Port', click the 'Driver' tab, and click the 'Driver File Details' button.
If there are no drivers listed, then remove the mouse from Device Manager, restart the computer, and let Windows recognize the mouse and reload its drivers.
If that does not work or if the drivers are already listed, then they may be corrupted, and you will have to reinstall them.
If the mouse is a standard PS/2 mouse, then the driver files are probably vmouse.vxd, msmouse.vxd, and mouse.drv.
These drivers can be extracted from a Windows ME installation CD, or possibly from a Restore CD that came with the system. But since the mouse works only in Safe mode and a CD-ROM drive will not work in Safe mode, then you will have to extract them onto a floppy on another system, and then copy them off the floppy to the C:\Windows\System folder in Safe mode on the problem system.
If it's a mouse that requires special drivers, then check the manuf's site for them.