I'm so sorry your beloved pet died! If possible, bury the body on your property. If you can't do that, you can call your pet's veterinarian for options. If you need help, call:
ACC Birmingham: 205-591-6522
Options to get rid of a dead dog body in Birmingham include burial on your property, burial at a Birmingham pet cemetary, or cremation. A Google search or your veterinarian should be able to tell you all of your options. If it's a smaller pet, such as a smaller dog, a dead cat, a hamster or gerbil, bird or reptile, you still have these same options, or you can actually dispose of the body with your normal trash disposal. You might not like that, but it's an option. You can wrap the animal un plastic and cloth.
If you want to keep the body fresh for a time before you can transport it for burial or cremation, you can wrap it up and put it in the freezer.