The pair became engaged in South Africa on the season finale of the ABC romance reality show on Monday night.
They appeared together on the following programme After The Final Rose, which was taped in advance and hosted by Chris Harrison, but have now said that they want to return to their normal lives.
In a statement issued by ABC, Brad said: "At long last, Emily and I can be open to the world and each other about our relationship. Though we are madly in love, it has been a bumpy couple of months for us and our priority right now is to focus on our relationship.
"We hope that you can respect our privacy as we return to normalcy and begin our off-screen lives together."
The Bachelor airs on ABC. The new season of The Bachelorette, starring Ashley Hebert, premieres on the network on May 23.
> The Bachelor recap: Finale
> The Bachelor recap: After The Final Rose