I presume that your current modem is connected to your pc through a USB connection. These commonly have a "dial-up" connection process, though you don't actually dial anything up.
You can replace your current set-up with an adsl modem route, which would be connected constantly whilst switched on.
You don't say how far from the phone socket you want to move it. If you are going to keep it in the same room and you could run a cable between the two, your best solution is a cable ethernet connection; otherwise, you can use wireless. Cable ethernet is much faster and, in the context of living in a flat, more secure. I would recommend it if possible.
However, these days it is little cheaper to buy a modem router without wireless than one with. However, pcWorld is not the cheapest supplier. Have a look at eBuyer: click here
This will do the same job for about £65 delivered, or with a cable about £5 more.
If you decide to go wireless in a flat, you would be well advised to make your wireless connection secure. Come back for advice on that, if you wish.