If a new, fully-charged battery is dead within one week, that is NOT likely to be a battery problem. You may have something in the vehicle constantly draining it. It could be a light that is on - inside you car, or maybe in the trunk where you don't notice it. An electric seat heater? Could even be bad diodes in the alternator that allow current to leak back when not charging. You need a proper diagnosis, and not a simple replacement of the battery. Doing that will only give you the same problem.
Finding a current user causing the leak is not nearly as simple as pulling a fuse to watch for a spark. Any item that draws enough current to make an obvious spark when you pull the fuse will be a BIG! current draw that would drain the battery in a few hours. If you are familiar with using it, a amp meter can help. You disconnect the GROUND cable at the smaller battery post, then connect the amp meter between that post and the cable clamp. That will show you the amount of current flowing in total, and for most cars with nothing operating, it might be a few milliamps for the security system or something. Then you can start pulling fuses one at a time and replacing. If pulling a particular fuse suddenly drops the amp reading, you have found one circuit that was using whatever that power drop was.
To check for bad back-leakage though the alternator, you need to disconnect the main output cable from the alternator. Be CAREFUL! - this cable is connected to the battery + terminal and is LIVE, so do NOT let it touch Ground. Now connect your amp meter from that cable to the alternator terminal. The current flow when NOT operating should be almost nothing.
Here's another cause that is rather common. The alternator (more precisely, some components inside it) may have partially failed so that it is not really re-charging the battery, even though a voltage reading says it should be. This can be tricky to diagnose for the backyard mechanic, and a garage needs to do a proper test of the alternator. In my city, there are some garages and parts shops that will do that test for you for free if you remove the alternator and bring it into the shop. Sure, they want to sell you a new one, so they may be a little biased, but if you trust your shop this is the way to test for a bad alternator. If you are not up to the removal task, most garages can do the test for you on the car for a fee.