I’m not familiar with streaming software…
The simplest way is to simply plug the P.A. mixer’s line-out into the soundcard’s line-in. You might need a Y-adapter if there is only one line-out and it’s going to a power amp.
The problem with that is, usually much of the music is acoustic, or there are separate instrument amps, so what you are hearing in the church is not exactly what you hear from the P.A. Even if there are choir mics, much of the choir-sound is not coming through the P.A. You also don’t get enough of the congregation through the P.A. So, simply recording the P.A. mix wont’ give you a good result.
To help with that you can get a 2nd mixer (and additional mics) for the music and congregation. You either run the P.A. mixer output into the 2nd mixer, or you can run the “split” the existing P.A. mics, running those mics into both the P.A. mixer and the recording-streaming mixer.
Or, you can get a high-end mixer with multiple buses (sub-mixes) to create separate P.A. and recording-streaming mixes.
It would be very helpful if you can have one guy (or gal ) running the live sound, and another with headphones running the recording-streaming mix.
What if I wanted to record 8 different channels and then adjust/mix them on the PC ?
You can get a multichannel interface ([u]example[/u]) and [u]DAW software[/u], but I don’t recommend it. Multi-track recording is great if you are going to spend a day, or a week, in post-production getting the mix just-right. Live mixing is a different animal.
One last thought… Computers are the must unreliable things we own! When there is no chance for “take two”, I strongly recommend a back-up device recording in parallel. (It can be a 2nd computer, a digital recorder, or even a VHS recorder.)