William Branham did not believe that illegitimate children could go in the rapture:
What William Branham taught
William Branham believed that a child born in sin could not be predestined to go in the rapture. He believed taht the person could be saved but not part of the Bride of Jesus Christ.
The concept that William Branham taught was that "hybrid" seed cannot produce "original seed". It produces death. This was one of the themes of his ministry. He ties "hybreeding" with "bastard" born children. And this all ties back to his concept of Serpent Seed. Effectively, William Branham believed that God regards natural seed lines above the Blood of Jesus Christ.
What the Bible teaches
The Bible teaches that when we are born into the Body of Christ, we become SONS and daughters of GOD.
- And if ye be Christ's, then are ye Abraham's seed, and heirs according to the promise.
A Christian is from a faith-seed line....not a natural seed line. BY FAITH - righteousness is granted to us.
- But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it— the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction: for all have sinned and fall short of the glory of God, and are justified by his grace as a gift, through the redemption that is in Christ Jesus, whom God put forward as a propitiation by his blood, to be received by faith.
If a person does not understand this, they do NOT understand the GOSPEL.
In other words, you might be the daughter of a prostitute, and not even know who your father is - and Jesus Christ can clean you - PURE and WHOLE.
- Yet you say, ‘Why should not the son suffer for the iniquity of the father?’ When the son has done what is just and right, and has been careful to observe all my statutes, he shall surely live. 20 The soul who sins shall die. The son shall not suffer for the iniquity of the father, nor the father suffer for the iniquity of the son. The righteousness of the righteous shall be upon himself, and the wickedness of the wicked shall be upon himself.
- But if a wicked person turns away from all his sins that he has committed and keeps all my statutes and does what is just and right, he shall surely live; he shall not die. 22 None of the transgressions that he has committed shall be remembered against him; for the righteousness that he has done he shall live. 23 Have I any pleasure in the death of the wicked, declares the Lord God, and not rather that he should turn from his way and live? 
It is clear that William Branham's theology was not Biblical.
THIS is the Gospel.
Where does this teaching lead?
IF this were true, message preachers should be honest with their congregations - and preach it. Because there are many people sitting in their churches who don't quality for the rapture. And they ought to have the courtesy of being informed that they don't qualify for the rapture. There are likely some message preachers that don't qualify either. Maybe they should have the 'non'-bride members leave, because they're not worthy. Why not conduct a poll of the congregation to weed out the illegitimate children? Wouldn't that be helpful?
At least then, people who were born out of marriage wouldn't have to keep striving for perfection. They're automatically 'foolish virgin', at best, so why keep them in the dark?
- Dear Brother Branham, if a baby is born of a--out of wedlock, can this child ever be saved or go in the rapture?
- Saved? Why sure, I believe the child could be... The child can't help what's been done. That is true. But I... About going in the rapture... Saved, I'd say, "Yes." But in the rapture it's a predestinated seed that goes in the rapture. And I can't believe that adultery was a predestinated seed. You understand? See? I believe the people can--the baby can be saved; it has no--no rights of its own. It's a awful act and things.
- But now, in the Old Testament when a baby was born a bastard child, it could not even enter the congregation of the Lord for ten generations, four hundred and something years, that curse was so bad. Just think of that. A innocent baby, its great, great, great, great, great, great, great, great, great grandchildren, just think, the baby's great, great, great, ten great grandchildren could not even come in the congregation of God. That's right.
- But you see, there was nothing there strong enough to forgive that sin. The blood of the Old Testament did not forgive sins; it covered sins. But the Blood of Jesus Christ divorces it. That's different now. When the Blood of Jesus Christ comes in, it's the different.
- Like the perverted vine, Jesus said it wasn't so at the beginning. And it isn't so at the beginning. God wanted every seed to bring its kind. And we hybreed anything; we ruin God's program.
- You take a flower. The original flower, a violet is blue, or white. Leave it alone, it'll go back to white. You have to keep breeding it up all the time. God makes things different. He likes it different. And hybreeding is such a dangerous and damnable thing till it ruins God's plan, it ruins the human race.
God even said in Deuteronomy that a bastard child would not even be permitted into the generation, into the congregation of the Lord for ten generations. That's how bad adultery is: ten generations: ten times forty: four hundred years before a hybrid...
- A woman that's married to a man, or a man married to a woman that'll cross the lines of holy wedlock, and bring in a illegitimate child, the generations will not come into the Kingdom of the Lord until ten generations. Deuteronomy 30--23:2.
- All right, it won't come in--a illegitimate child, he or his generation, or his generation, or his generation; even though they're trying like Esau, weeping to find a place, it takes ten generations. Forty years is a generation. Can you see this damnable, corruptible world is ready for judgment? So evil, cross up--cross up what God has joined together.
- A woman live unclean to her husband; husband live unclean to his wife and bring a hybrid child. Oh, some woman'd say, "Oh, isn't he beautiful, great big tall fellow." That's the same thing maybe Eve thought. She brought a hybrid child, and the hybrid child was Cain; and through Cain come giants. It's always been the cursed of the Lord: hybrid. Stay away from it: hybrid religion. Don't mix it.
- Galatians 3:29
- Romans 3: 21
- Ezekiel 18:19-24
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