Daniel Wani said Meriam Ibrahim was well when he saw her on Wednesday a day after she gave birth in prison.
According to Islamic law, she can nurse her baby daughter for two years before the sentence is carried out.
Born to a Muslim father, she married Mr Wani, a Christian, in 2011.
Sudan has a majority Muslim population. Islamic law has been in force there since the 1980s.
Even though Meriam Yehya Ibrahim Ishag, 27, was brought up as an Orthodox Christian, the authorities consider her to be a Muslim.
At her trial earlier in May in the capital, Khartoum, the judge also sentenced her to 100 lashes for adultery because her marriage to a Christian man was not valid under Islamic law.
'Love story'
Mr Wani, who is a US citizen, said he was delighted to see his new daughter - and mother and baby were both doing well.
"It's very incredible. I'm so happy," he told the BBC's Newsday programme.
But he said he was most concerned about his 20-month-old son who has been living with his mother in prison since February.
The judge ruled that Mr Wani was not allowed custody of the boy as the marriage was not valid.
"His attitude has changed a lot," Mr Wani said of his son.
"He used to be a happy boy. When I went there, he just looked at me. No smile," he said.
Meriam Yehya Ibrahim Ishag will be allowed to nurse her daughter for up to two years
"Sometimes really he is in a bad mood. Every time when I went there, he just wants to come home with me."
Born in South Sudan before it became independent from Sudan, Mr Wani went to the US in 1998 at the height of the civil war,
He met Ms Ibrahim in 2011 on a visit to Sudan and they were married at the main church in Khartoum.
"There is a love story between them," their lawyer Elshareef Ali told the BBC.
Mr Wani, who is in a wheelchair, said he was angry about his wife's imprisonment.
She had to give birth with heavy chains on her legs, although when he saw her in the office of the prison, her shackles were removed, he said.
Ms Ibrahim was raised as an Orthodox Christian, her mother's religion, because her father, a Muslim, was reportedly absent during her childhood.
Meriam Yehya Ibrahim Ishag said that as she was brought up a Christian, she had not committed apostasy
Mr Wani said it was his wife's right to choose her own religion.
"She grew up... with her mother, went to the church and I don't think that means that she converted from Islam to Christianity."
According to their lawyer, Mr Wani and his wife were first arrested in September 2013 for adultery - and were allowed out on bail.
The court added the charge of apostasy in February 2014 when Ms Ibrahim said she was a Christian. She was then taken into custody.
There has been international condemnation of the death sentence.
Correspondents say they are rarely carried out in Sudan.
The sentence of 100 lashes for adultery will reportedly be carried out when she has recovered from giving birth.
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