No you don't remove the whole rear end. Although that is an option of course if you can find one cheap enough.
To remove the diff centre, (the bit that has been welded) you first need to pull the axles out as the ends of the axles sit inside the centre. Once they are out, you can unbolt the centre from the housing and lift it out.
The axles are held in at the wheel end by 4 bolts. You undo these and pull the axle out. Easy. Except that at the wheel end is a bearing that is usually a pretty tight fit inside the housing. You use the spare wheel es a slide hammer to pull it out. Sometimes it's hard work, sometimes it pops out easily enough.
Put the wheel on the wheel studs backwards, put the 4 wheel nuts on backwards but loosely (too loosely and you get a face full of wheel) then slam the wheel out towards you till the axle gives.