Boot to the GRUB menu. Then, press <e> (for edit) before Linux has a chance to boot.
Using your keyboard arrow keys, move the cursor down a line or so, and press <e> (for edit) again, you should now be on the kernel line, and press <End> to go out to the very end of that line. Now hit the spacebar once, if necessary, to add a space, and then add one of the folllowing:
Then press <Enter>, and you're back at the kernel line, press <b> (for boot) and the computer will proceed to boot to a root prompt.
At this point, the root file system is still mounted read-only, so enter the following command:
mount -n -o remount,rw /
Enter your new password. Retype if asked to confirm.
Then restart the machine by typing:
Now you should be able to login with your new root password as required.