Technically you should be able to. Just get a Parallel ATA (PATA) drive and it should just plug right in. You may also see these labeled as ATA or some version of ATA100, ATA133, etc. A number of years ago, everyone started moving to serial ATA (SATA) and these are going to be the drives you will find everywhere now and that is NOT what you want. Here's a link from Newegg of PATA drives:
http://www.newegg.com/Product/ProductList.aspx?Submit=ENE&N=100007603%20600...
Now the question is why? Depending on what your use case is for this drive and enclosure, you may get better bang for your buck just getting all new (something with SATA drive(s) inside). Just something to consider.
Disclaimer: You really need to check the details of the Lacie enclosure to know what it can take. You didn't post the exact model of the enclosure, so my answer above is based on what I've generally seen across other similar enclosures.