Who is your current ISP. Is it Broadband (monthly subscription) or Dialup PAYG.
If it is PAYG just stop using it and start another account with another ISP.
If it is BB you have two options. 1. wait until your contract ends and don't renew. Then choose another BB ISP. or 2. Transfer it from your existing ISP to the new ISP (which can be fraught with problems). From what I have read you need to get agreement from your current ISP to allow the transfer. You will then be given a MAC number that you pass on to your new ISP. But I don't know any more than that.
Either way (with Broadband) you will be "offline" for a period of time.
I am on Broaband but still have an analogue modem and a PAYG dialup account active as a backup.