You don't mention what current source IC you plan to use. However, you will have to build your own circuit to use an IC. The IC itself will need a power source. You would need to have a relay to turn the output on and off. Then you would have to build some circuit to allow communications between the IC circuit and the PC. Then you would have to write a communications program DLL that Labview could call. This is a MAJOR undertaking and will not be easy. I suggest you stick with the cumbersome machine.
If you still want to proceede with your IC and you won't be varying the current, you could build your circuit and connect the output to a relay. Labview can then control the relay either throught the parallel port (see Labview example on parallel port), or through a DIO that you would have to buy.