Heather's father Mark Mills, who has had a five-year-long feud with his famous daughter, is now expected to launch a legal battle in order to gain access to his new granddaughter.
Mills - who devotes her energies to anti-landmine campaigns - had feared she would be unable to have children following complications when she was younger.
She had suffered two ectopic pregnancies a number of years before meeting Sir Paul.
Mills recently described the tearful moment when she revealed she was carrying their longed-for child.
She told US chat host Larry King: "I went upstairs and he was in a meeting. I kept walking in and out of the meeting because I had done the test and it was positive.
"He kept looking at me, thinking 'She's not normally like this'. I was going 'are you finished yet? Are you finished yet?'.
"Then he came out and I showed him the test and he just went 'Oh'. We both started crying and it was just a miracle."
The couple's last public appearance was at the UK premiere screening of a tribute concert for George Harrison in London earlier this month. She wore a long burgundy crushed velvet gown which showed off her bump.
SEND your congratulations to Macca, Heather and their new baby using the special messageboard by clicking here...
SIR Paul is one of the nominees in the Scouseology Awards 2003 - click here to vote
