When I looked up the chimney in the house we just moved into, found the throat damper doesn't stay closed. It's pretty badly corroded and should probably be replaced. It is one of the type that has a lever to close. How difficult a job is it to replace?

Best to call a professional. Those are usually attached near the smoke shelf in the chimney. They can be tuff to bust out and reinstall a new one.

Now you might get lucky and find just a damper if the original company is still in business. The lever and screw type dampers are about the same it is just the attaching hardware that is different.

You will have to remove the swing damper only and go to different brick/block suppliers to see if they can match it up.