Most of the natural pectin in an apple is in the apple's core, so it's important to include them in this recipe. Also, use tarter and smaller apples as they tend to have more pectin per ounce of apple.
Combine apple pieces with water and mint in a large pan. Bring water to a boil then reduce heat and cook 20 minutes, until apples are soft. Add vinegar, return to boil. Simmer covered, 5 more minutes.
Use a potato masher to mash up the apple pieces to the consistency of thin apple sauce. Spoon the apple pulp into a muslin cloth (or a couple layers of cheesecloth) or a large, fine mesh sieve, suspended over a large bowl. Leave to strain for several hours. Do not squeeze. Note that if your mash is too thick, you can add 1/2 a cup to a cup more of water to it. You should have 4 to 5 cups of resulting juice.
Measure the juice, then pour into a large pot. Add the sugar. Heat gently, stirring to make sure the sugar gets dissolved and doesn't stick to the bottom of the pan and burn.
Bring to a boil. Cook for 10-15 minutes, using a metal spoon to skim off the surface scum. Continue to boil until a candy thermometer shows that the temperature has reached 8-10°F above the boiling point at your altitude (boiling point is 212°F at sea level, so at sea level the temperature should read 220-222°F). Additional time needed for cooking can be anywhere from 10 minutes to an hour or longer, depending on the amount of water, sugar, and apple pectin in the mix.
To test for readiness, put a half teaspoonful of the jelly on a chilled plate. Allow the jelly to cool a few seconds, then push it with your fingertip. If it wrinkles up, it's ready.
Pour into sterilized canning jars to within 1/4 from the top and seal. Makes approximately 4 8-ounce jars.