ALFA USB wifi adapter

I have a ALFA USB adapter. It works just fine. This is the only internet connection I can have.
I need to connect a dongle. Serial to ethernet adapter system to the internet.
This adapter is connected to a unit via the serial port that needs to be connected to the internet.
It also will need to be configured with my computer using WIZnet software.
Is there any way I can connect this dongle to the ethernet jack on the rear of my computer and have it connected to the internet?