Verses 32, 33: "And they spake unto him the word of the Lord, and to all that were in his house.
And he took them the same hour of the night, and washed their stripes; and was baptized, he and all his, straightway."
We see that the jailer's family too believed and all of them got baptized in water before daybreak. Take that as proof that God fulfills the promises He makes, provided we fulfill the part we need to. In this case, our part is to believe on the Lord Jesus Christ (believing that he is the Son of God who died for the forgiveness of our sin and who came back to life to give us everlasting life). So if you are born again, you have already fulfilled your part. So God fulfilled the first part of His promise - He saved you. Now He will also save your entire household.
"Thou wilt keep him in perfect peace, whose mind is stayed on thee: because he trusteth in thee." - Isaiah 26:3