Question:

what is the ruling for adopting a baby in Islam? 

Answer:

Adopting is to take somone else's child as your son, that's haram in Islam.
However, caring for a child or orphan (but not claiming to be his father) is not haram, rather caring for him is very well recommended. But the caregiver should tell that child that he is not the father.

Proofs on this are verses from the Qur'an such as:

The ayah talks about Zaid whom the Prophet took as a son first, then when the Ayah was revealed, he showed the ruling that adoption is probhibited.

Also this Ayah: {مَّا كَانَ مُحَمَّدٌ أَبَا أَحَدٍ مِّن رِّجَالِكُمْ وَلَكِن رَّسُولَ اللَّهِ وَخَاتَمَ النَّبِيِّينَ}
So it shows that Prophet Muhammad is not the father of any of your sons, rather he is the messenger of Allah, and affirms that adoption is prohibited.

And Allah knows best.