Combine juice and 1 cup water in a wide, heavy-bottom pot. Roughly chop peaches; add to pot. Bring to a boil over medium heat. Reduce heat to low and simmer, stirring occasionally, until peaches are tender and just beginning to break down, 10 to 30 minutes, depending on ripeness. Slightly crush mixture with a masher, leaving some large chunks.
Add sugar and cinnamon; cook, stirring constantly, until sugar has dissolved, about 2 minutes. Increase heat to high and boil rapidly, stirring constantly, until thickened and at the gel stage (see below), 20 to 30 minutes. Skim off any foam from the top and discard.
Ladle jam into half-pint jars, leaving 1/4 inch of headspace. Process for 10 minutes using the boiling-water method (see below).
How to test for the gel stage:
When preserves hit that magical jammy consistency--spreadable but not runny--it's considered the gel stage. Jam sets at 220ºF. Use a candy thermometer or these visual cues.
Watch the jam as it cooks. Bubbles will increase in size as the gel stage nears.
Lift some jam with a wooden spoon and tip it back into the pot. When it runs off in drips that thicken into a flat sheet before coming off the spoon, it is a sign that the jam has reached the gel stage.
Drip a bit of hot jam onto a frozen plate. Give it a minute, then push it with your finger. If the cool jam wrinkles as you nudge it, your jam has reached the gel stage.
Don't be intimidated: Getting the right seal to keep food fresh is as easy as boiling water.
STEP Put the rack in a large pot. Arrange empty jars, open side up, in a single layer. (Arrange a full layer, even if you are planning to use only a few, to keep jars from tipping over.) Add cold water to the pot until there is at least 2 inches of water above the jars' rims. Set the pot over high heat, cover it and bring water to a boil. Put lids, white side down, in a large bowl.
Step Prepare recipe as directed up until the ladling step (keep hot until ready to place in jars). Reduce heat under the pot so water is hot but not boiling.
Step Using canning tongs, remove a jar of hot water from the pot. Pour it over the lids in the bowl. Set the jar on a clean kitchen towel. Remove 2 more jars, pouring the water back into the pot, and set them on the towel.
Step Insert the funnel into the first jar. Carefully ladle the hot food into the funnel, keeping the headspace--the distance from the top of the mixture to the top of the jar--indicated in the recipe.
Step Swipe a bubble tool between the mixture and the inside of the jar to release any air bubbles, adding more of the mixture if necessary to achieve the proper headspace. Repeat with remaining food and jars.
Step Use a clean kitchen towel dipped in hot water to carefully clean the rim of each jar. Use soft-tipped tongs, a lid lifter or your fingers to retrieve a lid from the bowl and center it on the jar. Screw the ring on the jar with your fingertips, taking care to make it tight enough to stay centered but not so tight that you won't get a vacuum seal. Repeat until you have filled the jars.
Step Use canning tongs to lower filled jars into hot water. Be sure jars are covered by 2 inches of water. Cover the pot and return to a boil. Begin counting your processing time when the water is boiling. When the time is up, turn off heat, remove pot lid and allow the jars to rest for 5 minutes.
Step Using canning tongs, remove the jars to a towel-covered surface. Let rest for 24 hours. To test the seals, remove the rings and gently push up on the lids. If they stay in place, the food is safe to store for up to a year in a cool, dark place. If the lids don't stay put, refrigerate the jars and eat the contents soon.