Having proved that the drive is physically faulty, the next step (after backing everything up!) is still to check whether the drive is within warranty. The laptop warranty and hard drive warranty aren't necessarily related. Dell often uses Hitachi drives, which may have 3-year warranties. (As you haven't mentioned the brand or model of drive, it's all guesswork.)

Dell themselves will simply sell you a drive at a mark-up, so going to them for a replacement would be a waste of money. Decide what size/brand/model of drive you want and get it wherever you can get the best deal. If you use your laptop a lot, it's probably worth going for a brand which gives you a good warranty, obviously.