He had claimed he was in his girlfriend's flat at the time of the infant's death.
But forensic evidence showed his presence at the location where the baby's body was found, wrapped in a bin liner, in a nearby brook.
"This is a case in which you are perhaps the only person who will ever know the precise truth about what happened to Troy," Mr Justice Gibbs told Smith at Birmingham crown court as he passed down the sentence.
Smith, who was born in Jamaica, was sentenced to an "indefinite" term in jail and will serve at least three years and 329 days before becoming eligible for parole.
"I am satisfied that you set out in the early hours of the morning with the object of doing something with or to Troy," the BBC quoted Mr Justice Gibbs as saying.
"Certainly this included secretly taking him from his cot. Your intention was to do something that you realised might cause him harm."
Smith believed he was Troy's father at the time of the baby's death. He received a permanent ban on working with children.
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