What you need is an FXO card for your PC. This is a PCI card with a phone socket in it, and it effectively works as a PSTN (public switched telephone network) phone on a card. You plug this into your wall socket just as you would a telephone.
Then you use PBX software (such as Asterisk) to mediate the calls. You would need a way to interact with the PBX, and this is usually done with VoIP, even if the FXO card and the headset are attached to the same machine. You would use a softphone that connects to the PBX over VoIP and then the PBX would make the ongoing call over PSTN.
You will still need a splitter. The broadband connection needs to connect to your ADSL line directly and to the FXO card via a filtered splitter.
The advantage of this, over just plugging a hardware phone into the broadband line with a splitter, is that you get to take advantage of the features of a PBX like voicemail, voicemail to email, fax to emails. The advantages increase when used in conjuntion with an outgoing VoIP service.