When they founded Harry’s as a shaving start-up nearly nine years ago, Andy Katz-Mayfield and Jeff Raider sought to shake up a decades-old business dominated by two giants in the field.

Now, they’re joining forces with one of them.

Edgewell Personal Care, the company that owns the Schick and Wilkinson razor brands as well as Hawaiian Tropic, plans to announce on Thursday that it will buy Harry’s for about $1.37 billion in stock and cash. The deal will lead to Mr. Katz-Mayfield and Mr. Raider running Edgewell’s operations in the United States.

It is the one of the largest recent examples an established business buying a younger, nimbler competitor born of the internet and predicated on reaching consumers in new ways. That has included deals like Unilever buying Dollar Shave Club, the other shaving start-up sensation, for $1 billion three years ago, as well as Walmart acquiring the online men’s wear purveyor Bonobos for about $310 million.

Harry’s sells razors, face washes and lotions, as well as the Flamingo line of women’s razors and waxes, directly to consumers over the internet, offers subscriptions (a certain number of blades delivered to your home each month) and has struck a collaboration with the clothing retailer J. Crew. Based in New York, it owns the German factory that makes its blades and has invested in other businesses like Hims, a start-up that sells hair-loss prevention products.