The pair were first spotted getting cosy at an SNL after-party on May 12, and since then have enjoyed a whirlwind courtship, with Davidson shocking fans last week when he unveiled two new tattoos in tribute to his famous girlfriend.
Davidson got a black bunny ear mask tattoo behind his left ear, and the initials 'AG' on his right thumb.
Grande liked several tweets congratulating her on the engagement and then tweeted a semi-cryptic, emoji-filled message after the news broke:
It's less than a month since Davidson confirmed he had split from his previous girlfriend, Larry David's daughter Cazzie. Grande broke up with her longtime boyfriend, rapper Mac Miller, in May.
Given the remarkable speed of their engagement, it seems not everyone is as excited as the happy couple about their news — while many fans have tweeted their congratulations, comedian Kathy Griffin directly warned Grande to "get a pre-nup":