Her panicked partner was trying to get her to hospital in Carmarthen while she was in the last stages of labour.
Armed response officers Darren Trevan and John Woodcock took control of the situation - and got the couple to the maternity ward just in time.
Mother, father - and their new baby son are all doing well - said Dyfed-Powys Police.
The two officers had initially been a little "surprised" when their vehicle was overtaken by a speeding car, and they soon pulled it over to question the occupants.
"It quickly became apparent that one of them, a heavily pregnant lady, was in the very final stages of labour," said a Dyfed-Powys Police official.
"A quick master-plan was devised where PC Woodcock drove the couple's vehicle to hospital, escorted by PC Trevan, leaving the driver to comfort his partner.
"Thankfully they all made it to the nearest hospital just in time for the birth of a lovely baby boy."
Armed officers who spotted a speeding car in west Wales discovered a mother-to-be about to give birth.