Prosecutors say he is looking at a prison sentence of 24-30 months. U.S. District Judge John Adams will sentence Tomasone at a date yet to be determined.
Tomasone was charged late last year and, at that time, pleaded not guilty.
He initially pleaded not guilty to conspiracy to commit honest services mail fraud and honest service fraud.
Southwest provides painting, roofing, waterproofing and paving services.
The indictment alleges Tomasone gave cash, gift cards and other items to Christopher Krause, who was treasurer of the Maple Heights School District, and former Maple Heights School Board member Santina "Sandy" Klimkowski.
Krause pleaded guilty July 1, 2010 and is awaiting sentencing. Klimkowski pleaded guilty Oct. 2, 2009 and is also awaiting sentencing.