David Sinclair, 34 admitted the culpable homicide of his son Joshua at the family home in East Kilbride, on December 6, 2012.
He was originally charged with murder, but admitted the reduced charge of killing his son, just days before he was due to go on trial.
The supermarket worker married his wife Kirsty, who is pregnant, on the anniversary of their son’s first birthday on July 9, 2013.
She originally stood by Sinclair, but has now said she can no longer support him.
At the High Court in Glasgow judge Lord Bannatyne told Sinclair: “You assaulted your baby son by shaking him, causing the injuries which killed him.
“Your wife believed in you. She now feels she had been misled by you and can no longer support you.”
The court heard Sinclair has problems with anger management. He showed no emotion as he was led away.
Defence counsel Ian Duguid QC said the case took so long to resolve because Sinclair could not accept his actions caused the baby’s death and medical investigations were undertaken to see if Joshua was suffering from a genetic defect that would explain his injuries.
The court previously heard that that on the day the child died, Sinclair and his now wife had gone swimming with Joshua and then shopping in East Kilbride. Later, Kirsty Sinclair went out to meet her mother and sister. Just 11 minutes later she received a phone call from Sinclair, who sounded “shocked”. He told her: “Joshua isn’t breathing properly, come home right away.”
He then hung up and dialled 999.
The infant was rushed to Hairmyres Hospital where he died.
Initially the cause of Joshua’s death was unascertained, but the case was reviewed by a team of medical experts who discovered evidence of trauma in the eyes and brain which could only have been caused non-accidentally, close to the time of death.
One of the experts, Dr Peter Richards, said: “This would be compatible with him having suffered a shaking injury involving handling at greater force than encountered in everyday life.”