Alexander Mackay's case was described by a sheriff as "one of the worst management failures I've ever seen".
Dundee Sheriff Court heard the G&D Pallets saw room in Dundee was in an "appalling state" when Michael Rice was injured.
Mackay, 65, will be sentenced on 2 May.
The court was told surgeons were only able to reattach one of Mr Rice's fingers, leaving him permanently disfigured and impaired.
Depute fiscal Shona McJannett told the court that the incident took place while Mr Rice was sawing wood at the company's warehouse.
Miss McJannett said Mr Rice was employed as a casual worker in 2010 and given "basic training".
He was then taken on again on a casual basis in 2011 and given no refresher training.
The prosecutor said: "At the time the premises were very busy and they were struggling for space in the saw room.
"There were pallets partially blocking one door, completely blocking the second door and stacked up throughout the room and around the saw.
Miss McJannett said Mr Rice was working alone in the saw room at the time of the incident.
Mr Rice had turned and tripped over a broken pallet that was lying near the saw bench where he was working.
The fiscal said: "As he fell, he held out his hands and his right hand came into contact with the running blade of the saw.
"The emergency stop button was damaged and didn't work, which Mr Watson said he was unaware of but other employees had known for some time."
Mackay, of Blairgowrie, admitted a charge under the Health and Safety at Work Act committed in December 2011.
Sheriff Alastair Brown said: "This is one of the worst management failures I've ever seen.
"The workplace was in an appalling state and resulted in significant injury to someone who was vulnerable.
"That being so there is a very serious possibility of a prison sentence being imposed."
A company manager has been warned he faces jail after a worker tripped over a broken pallet and fell into a moving sawblade, slicing off three of his fingers.