Hi Andy
From the spec of your computer, everything is very good bar the graphics card (god only knows why retailers put good parts into a computer, but use a next to rubbish graphics card or even intergrated graphics card!).
Several things we will need to know. First off as you identified correctly is the power supply 'box'. What we idealy need to know is the wattage rating it has. This should be found in the manual that came with the computer or, as you say as you are upgrading the graphics card, take off the side panel of the computer, and look at the 'box' at the top of the case. It will have a label reading several numbers on it with something like 'Total voltage output' and a value next to it. Post that if you can

. I heavily suspect that it might be a generic 300watt power supply but have no fear! we only need to know the rating at this stage.
Next we will need to know is what motherboard you have. Now, this might not be obvious off the bat but your manual again should say what motherboard you have. If not and since this I think this is a pre built computer, would it be possible to give us a website link to where you bought it? Otherwise, as you say you purchased it just a month a go, it probably will have a PCI-e x16 slot, which is where the graphics card will go. (As a last resort you could look on the motherboard for a label which generally tells you what make/type of board it is).
Once we know these two things, you can consider what graphics card to buy

. Now, if you do only have a 300watt power supply, as mentioned earlier, a Radeon 4670 will work wonders for you

. It only takes the power it needs from the slot its in, so no additional connectors are required. The 4670 can usually be bought for around £60 online.
However, if you have a higher rated power supply buy a reputable maker, you can consider buying an even better graphics card such as the Radeon 4850, Nvidia 8800GT etc. I would leave that decision until we know:
- What power supply you have (i.e. the voltage rating and the Make of it if possible)
- What motherboard you have (I'm pretty confident it will have a PCI-e x16 slot regardless anyway considering its only 1 month old

).
But ultimately, if you purchased the computer from a shop retailer, either online or instore, would it be possible to give a website link to it please so we can look at the list of specifications?

.
Graham