Fill a plastic bag with moist sphagnum moss. Place the pawpaw seeds in the bag and cover them with the moss. Seal the bag to keep moisture in.
Store the pawpaw seeds in a refrigerator set between 32 and 40 degrees Fahrenheit. Check the moisture of the moss periodically. Add a few drops of water to the moss if it dries out.
Remove the pawpaw seeds from the refrigerator after 90 to 120 days. Prepare one 3-gallon container for each pawpaw seed you want to plant. Fill the containers with potting mix with a pH between 5.5 and 7.0. Cover the pawpaw seeds with an inch of soil.
Set the pawpaw seed pots outdoors when temperatures reach 75 to 85 degrees Fahrenheit. Alternatively, put the pawpaw seed pots on a heat mat indoors if outdoor temperatures are not warm enough.
Wait for approximately 50 to 60 days to see a pawpaw seedling emerge. Transplant the seedling to a well-drained spot in the garden when spring arrives.