22-month-old Mitchell was found dead at home in September 2005.
In previous court hearings she was accused of giving the baby the drug to keep him quiet.
Teeside crown court was told by the prosecution that the case was tragic and no "wickedness" was alleged on her part.
A previous trial was halted for further forensic testing.
The baby's father, Mark Bate, 24, Hartlepool, was cleared of manslaughter last month after the prosecution offered no evidence against him.
Commenting on the case, detective chief inspector Gordon Lang of Cleveland police said that it was "appalling and tragic".
"[Mitchell] should have been cherished, loved and looked after but the amount of methadone that he had consumed would have caused serious problems to an adult user," he added.
"It had a profound effect on the small team of officers who were involved in investigating this case."
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