First, there should never be a time when you have both an integrated sound card and an add on sound card working at the same time. The fact that they aren't is the normal way of things.
Second, one of two things is true however. If when you insert an add on sound card into your motherboard and you have a plug and play bios then the integrated sound card should be disabled automatically and the add on sound card should be then enabled by default OR if you do not have a plug and play bios then either before or after you insert the add on sound card into your mother board then you have to go into your bios settings (on boot up press DEL or F2 or some such thing) and disable the integrated sound system. Now since you have the add on sound card inserted into your mother board and the integrated sound system isn't disabled (as evidenced by the fact that it is still producing sound) and your add on sound card isn't working we can therefore assume you DO NOT have a plug and play bios and MUST go into your system bios and disable the integrated sound system.
Third, are you sure the add on sound card is compatible with linux?