You can propagate by a couple colonal methods:
Air-layering - will give you a clone of the parent tree on it's own roots, (so will just about any of the layering or cutting methods)

Grafting is your other option. Best choice if you want to impart characteristics of the rootstock into the resulting tree, (tree vigor, dwarfing, semi-dwarfing, wet soil tolerance, drought tolerance, early fruit ripening, etc; all can be controlled via the root stock you graft the scion wood onto). The rootstock should have no impact on the quality of the fruit.

Planting from seed will produce a similar tree, but the fruit charateristics may not be as good and you'll have to wait for several years, (5 - 7, I am unfamilar with the maturity rates on apricots, so this is a guess); before you'll know if the fruit is as good as, better than, or worse than the parent tree.

So if you want to get more fruiting trees faster, the best choice is grafting.

I have been searching for the same information, since the main part of my huge old apricot tree blew down a couple of days ago. Since the tree has the biggest, best apricots I have ever tasted, I would like to clone it before my husband takes a chainsaw to the remnants of the tree.

Does anyone know if air layering will work this time of the year?

Thanks to the last commenters great information! I have one small thing to add. In my search, I have read that apricots are about the fastest tree to fruit and sometimes you will get some fruit in 3 years. So if you do try some seedlings, you would know relatively quickly if they are worth keeping. I have seen some relatively poor fruit from seedlings, though. It is a gamble for sure.

Good luck.

Apricot grafts easily to the appropriate rootstock. I grafted mine to Myrobalan plum seedling stock. Your planting location may call for a different stock so check locally. Al