If you want to grow tasty, juicy apricots in your own backyard, your best bet is buy a tree from a nursery. These plants are bred to produce good fruit and have already passed the testy seed germination stage. Grocery store apricot seeds sometimes won't sprout into seedlings, and those that do make it to maturity don't always bear high-quality fruit. If you still want to try growing a tree from stone, improve your chances by using seeds from organically grown apricots.
Remove the stone from a ripe apricot and soak in a bowl of water for one day.
Wrap the saturated stone in a piece of damp paper towel, then roll up the paper-towelled bundle in a layer of cellophane. Keep the seed in your refrigerator for a month, storing the stone at about 3.89 degrees C (-15.6 degrees C C).
Assess outdoor weather conditions. Your seedling will grow best once the danger of frost has passed and when days reach temperatures of 20.0 to 25.0 degrees C (-6.66 to -3.88 degrees C C). If weather conditions aren't ideal for outdoor planting, grow your seedling indoors until temperatures warm up. Place the stone about an inch from the soil's surface in your indoor planter or backyard garden. See that your developing tree gets lots of sun. Water once weekly, letting the soil dry out between waterings.
Wait for your new tree to sprout and make sure it gets the same amount of water and sunlight as it grows. If you planted your tree in a pot, transplant it outdoors as soon as the weather permits. The University of Wisconsin Extension advises digging a hole deep enough to accommodate all of the roots without bending or pruning them.
Try preparing and planting a few apricot stones at once. Some might not work at all, and some stones will produce healthier seedlings than others. Planting a few at once maximises your chances of getting a mature, healthy tree as a result of your efforts.