Verified answer

He promises it immediately.

without thinking...

I said yes without thinking when he proposed.

一口答應 = No Problem

他一口就答應了 = He has no problem with it.

He said OK right away. OK已經快變成世界語言了right away 馬上

He said yes without hesitation.

他毫不遲疑的答應

正確翻譯應該為 He agreed to it at once.