noun
DUBIOUS : Sitting in the next seat was a young woman who was having a baby.
GOOD : Sitting in the next seat was a young woman who was expecting a baby.
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have a baby = give birth to a baby: 'She's worried about having the baby at home in case there are complications.'
be expecting a baby = be pregnant: 'Janet is expecting another baby.'
Note that be having a baby can be used to mean 'be pregnant' but this should be avoided when there is a danger of confusion.