I purchased a brand new house back in April 2002. The house has two floors but only one air conditioning unit to power central air throughout both floors. It's not possible to add a second one to power the second floor. Do I have any alternatives to economically cool both floors? Currently, the first floor gets very cold and the second does not so I have to keep it colder than it needs to be to compensate.

Sounds like a balancing problem, but residential air conditioning seldom has balancing capability built in, so you may need to replace some ductwork. Basically, you need to deliver more air to the second floor, and less to the first floor. If you have adjustable diffusers, you could try partially closing the diffusers on the first floor, in the hope that more air will be forced to the second floor. butif the duct sizes aren't right, there is only so much that can accomplish.

If the A/C unit is sized correctly, then it's an air delivery problem.

Two options.
You can disconnect the ducts to the second floor and construct a new duct supply system in the basement to feed them only.
This is expensive and you would need to re-size the current system as the air ducts left would not be able to handle the flow required with what ducts were left.

Or you could properly re-balance the current system and add an additional return duct, which my guess is undersized anyway as most are.

You need to have a return near the base of the stairs to pull the cold air that falls down the stair case while the system is running. Not having this is the reason for the first floor being so cold. All cold air drops and fills the first floor. You need to capture this so it does not overcool the first floor and push it back up to 2nd floor level. You also need a cold air return duct near the ceiling on the 2nd floor. This pulls the heat off the ceiling and helps the cold air fill the upper rooms.

Using paddle fans is also a good option. This helps the air move around the room more thus mixing it better. They tend to lower the cost of the AC operation as the moving air helps cool as well.