The wallet-size case, cleverly designed by the drug company to fit the hypodermic needles and the tiny vials of hormones, sits on her kitchen counter, the only thing Leah really sees when she makes her morning coffee. The path she’ll start down in three days is scary, lonely, and a bit embarrassing—she had to ask a girlfriend to come over every morning to plunge a needle into her buttock—but also exciting. And she is sustained by an astonishing thought: I am freeing myself from the tyranny of the expiration date.

Leah, a willowy 35-year-old media-company executive, is accustomed to taking chances; that’s how she came to rise so high professionally at her age. But this one is big. She’s about to join the vanguard of what may be the most significant social change since the advent of the Pill, taking a technological step that in just the past year has emerged as a real option for women. She will spend at least $15,000—perhaps twice that—to freeze as many eggs as science can help her produce in a month or two and keep them in a cryogenic vat indefinitely. No one is officially tracking the numbers yet, but fertility experts say she is among the several thousand American women this year who are expected to opt for egg freezing, known technically as oocyte cryopreservation. “The science of egg freezing is finally at a place where it’s making a real difference in people’s lives and futures,” says Jamie Grifo, M.D., program director of the Fertility Center at New York University. “We’ve been working toward this point for years, and sometimes it’s hard to believe we’re here.”

Leah never thought she’d need to do this. The plan was to be married by now. But her on-and-off boyfriend, a gourmet-food importer, just isn’t headed in the same direction as she. He’s adorable but not serious. He smokes too much pot. Her boss at her last job had brought up egg freezing a couple of years before, after he and his wife, in their late 30s, had had...