Ten others have been spared a hanging, their sentences commuted to life imprisonment
The rulings were made by five judges headed by Chief Justice Ernest Sakala on Thursday.
Although the southern African country's High Court had already found the soldiers guilty of treason, they immediately appealed to the Supreme Court - arguing they had been caught up in the coup unintentionally.
Under Zambian law, the president - currently Levy Mwanawasa - has the final say on whether the death penalty should be carried out in each case.
In the aftermath of the bid to topple then president Frederick Chiluba, 59 soldiers were arrested but the High Court acquitted some while others died during the time their case was in court.
Zambia's founding president Kenneth Kaunda was also arrested in connection with the same coup attempt but was released without trial.
Chiluba, who was head of state from 1991 to 2001, is currently on trial on corruption charges - accused of stealing $45 million from the state.