First, we have to assume that you actually have an IDE port on the mobo to connect to. You will need a power supply connection (a 4-pin Molex) from the PSU, and a wide ribbon cable for data. The ribbon cable should have three connectors on it - one blue one at one end, one black one at the other end, and a grey one in the middle. Each connector will have two rows of 20 holes each, and one of those holes will be blocked off. That way you can only plug it into a socket one way. The cable should have 80 wires in it, even though that's a 40-contact connector on the ends. To check, start counting wires across. When you get to 20, you should be only ¼ of the way across the ribbon.
Since this is the only device on the IDE port, it MUST be set to be the port's Master. To do this, look at the diagrams on the HDD itself, and set the jumper on the sets of pin pairs on the back edge as it says. Do NOT use a diagram from another unit- there is no universal standard for this. If your unit has different settings for Master with No Slave, and Master with Slave Present, set accordingly.
Mount the HDD physically in your case. Connect power. Take the wide ribbon cable and connect the BLUE end to the mobo port. Then connect the BLACK one on the END to the HDD's data port. The Grey middle connector will be unused. Close up the case and boot directly into BIOS Setup.
Check on one of the first pages for an IDE device. Make sure the IDE port is Enabled, as others have said. Your BIOS should find a Master device present on the Primary IDE port, and its size should be about 60 GB, as you say, so check that to be sure the unit was detected correctly. If it is not, check near the end of your BIOS's menu system for a way to force detection of an IDE drive - that process might be better than the automatic one. If your machine cannot recognize the HDD with it set up this way, there's a good chance the old drive is dead.
By the way, IF your mobo has two IDE ports and you happen to have connected the ribbon cable to the Secondary port and it is working, that's quite all right. It really does not matter which IDE port you use. In either case, the only storage device on an IDE port MUST be set to be Master.
If your BIOS can detect the old drive properly you're good to go. One note: IF you had to change any BIOS Setting (like Enabling the IDE port), make sure to SAVE and EXIT so your setting are permanent. If the drive was detected, it may show up in Windows' My Computer if it contains data. If it is empty with no Partition, you may ave to use Disk Management to Initialize the old unit before use.
Let us now how it goes.