The incident took place La Villa de Don Fadrique, a village of around 4,000 people near the city of Toledo.
"She killed her baby in the cemetery chapel," a police source said, adding that the woman had apparently slit the child's throat.
A knife was found at the scene.
The 37-year-old woman, who also has another child, was arrested at the village church where she sought refuge after the attack.
Local residents called police after they saw the woman "crying out and covered in blood" at the cemetery just before 8:00 am (0600 GMT), the mayor of town, Jaime Santos Simon, told AFP.
Local media cited neighbours as saying she suffered from mental illness. The reports said the woman screamed that she "had the devil inside her body" as she entered the church.
While police were chasing the woman, other officers found the body of her three-month-old baby at the cemetery chapel, the mayor added.
"Nothing led us to think that she could do something like this," the mayor said, adding that the woman may have suffered from postnatal depression.
The town declared one day of official mourning in honour of the baby.
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