Katharina Katit-Staheli, 40, fled her home in Switzerland with son Dylan, who suffered from hydrocephalus, also known as water on the brain, but had been captured by Police in Spain
Officers had taken her to hospital where it is believed she managed to get hold of a knife which she used to stab the child.
The horrific incident happened just hours after the pair had been tracked down to the town of Torrevieja near Alicante by officers searching for them with an international arrest warrant.
Spanish police had put out a public appeal for information on their whereabouts just 48 hours earlier after the child's father reported them missing.
Guardia Civil officers took him to hospital after discovering them in a shopping centre in Torrevieja around 7pm yesterday/on Wednesday.
Dylan was attacked just before midnight at the town hospital.
An early report said he had been smothered before the mother - who no longer had legal custody of the child - was stopped as she slashed her throat with the same knife.
But a police source today confirmed the child had been stabbed to death - although he said a judicial secrecy order prevented him from giving out more details.
It was not immediately clear if the woman was under police guard when the incident occurred or if a police officer was keeping watch on the room mum and son were in.
The police source, who asked not to be named, said: 'I can confirm the child we were searching for has been killed in hospital and his mother - who is the sole suspect - subsequently tried to commit suicide.
'The child was stabbed to death. There's obviously going to be an investigation now into what happened because she had only been tracked down and arrested hours earlier following an international appeal.'
Police in Zurich had offered a reward of £2080 for information on Katharina and Dylan's whereabouts.
They were last seen on December 24 and are believed to have left Switzerland in a black Jeep Gran Cherokee.
A court spokesman said: 'A judge is investigating the circumstances surrounding the violent death of a 10-month-old baby discovered in Torrevieja with its mother who was the subject of an international arrest warrant.
'The mother of the child remains in hospital and is under police guard.'