Is it possible to install a second HDD in this situation?

As long as you correctly jumper the drives to have the old one as slave, sure no issues. If you want to really avoid any issues, buy an external case for the old drive.

So what you're saying is as long as I make the main drive "master" and the extra "slave", there won't be any problems?

And what do you mean by external case? I wasn't aware of that as an option.

Watch out for misinformation!

The terms "Master" and "Slave" are very specifically used to identify TWO drives that are sharing one IDE port and cable. They are useful ONLY in that situation. There are NO Master or Slave units on SATA ports (they only have one device per port). NO computer has a Master Drive and a whole bunch of subservient Slaves!

These days we DO talk about a Boot Drive for the computer, plus other drives that are used for data but not usually for booting. (It certainly is possible to have a system with more than one boot drive if it's arranged properly, but that is not your issue here.)

So, you are getting a working new computer with a SATA HDD, no doubt. Do NOT try to play with its jumpers. You also have you old HDD, which may be SATA, in which case you again should NOT change its jumpers. Or, it may be an IDE drive type, in which case you will need an IDE port and cable to connect it in the new computer. If you are doing that and the old HDD is the ONLY unit on that IDE port and cable, then its jumper MUST be set to the Master position. Any IDE port MUST have a Master to operate.

I said do not change jumpers on SATA units. There is one valid exception to that rule, probably NOT in your case. On SOME SATA drives a jumper can be installed on a particular pin air to force it to slow down to the original SATA 1.5 Gb/s communication speed. That can be used IF you have an older mobo with original SATA controllers. Otherwise you don't need to. I will warn you that SOME SATA drives use a jumper on pins to activate a special operating mode used with servers with special controllers. If this mode is set in a "normal" mobo system, the HDD will appear dead! So you can cause trouble playing with SATA jumpers. They are NEVER for setting Master and Slave because there are none in SATA!

Be aware that, if you have games and other applications software on the old drive, merely mounting that drive in the new computer won't necessarily make them work right. They are not "installed" when you do that, and the Windows OS in the new system will not have all the details for them to run smoothly. You may have to look around for info on how to get Windows to recognize such software and use it properly.

External case something like http://www.newegg.com/Product/Product.aspx?Item=N82E168....

You will need to check what connection your drive has, SATA or IDE and get the proper box. IDE will look like about a 1.5-2 inch wide connector with a bunch of pins, SATA looks like a 1/2 inch plastic connector with the data connectors on the bottom of it.

Then you can keep the old drive in the case and can even move it from one PC to another, and not worry about setting up the 2 drives in the PC case.