I'd start the seeds indoors, Benny to get them going so that they're a decent size before you put them outside. Good size transplants will tolerate light frost if they've been hardened off and given a chance to establish before the cold hits. But seedlings would probably not survive. Dill seedlings are very delicate and take a while before they're of a size to handle without damage. If you have a cool, bright windowsill you can keep them going nicely for a month or so. (it will take the seeds maybe a week to germinate). Or you could check your local Home Depot or Lowe's store for starts some time next month.
Knowing that high desert weather, you can have frost and even light snow right into May, right? If you created a little area you could cover with a light blanket or frost cloth, I'd think you could get your transplants out towards the end of March, but you'd need to give them some protection if a late cold snap comes along. Even cardboard boxes inverted over the plants will protect for a night or two.