Okay... so, although the firewire interface was not specified, it probably doesn't matter. Most interfaces will have at least one or two XLR inputs for microphones, and then Line-in balanced inputs.
All you need to do is run the balanced, 1/4" output from the UA 610 to one of the 1/4" balanced inputs on the firewire interface, using a balanced TRS 1/4" cable (looks like a guitar cable, but with "stereo" ends).
You shouldn't need a mixer in this chain at all.
Now, if you are trying to record more inputs than you have on the firewire interface, you would use the mixer to mix those inputs to a stereo track, and run the stereo output of the board into two inputs on the firewire interface.
After that you wouldn't be able to remix what you recorded.
So, I would assume in this instance you are trying to JUST record vocals. In that case, do what I suggested first. The mic preamp will boost the signal of your mic to the point that it can be recorded by any type of recorder that accepts line-inputs, which is a lot (most?) of them.
In regard to your question about mixer mic pre's:
If you were to plug your mic into the UA, then run its line out to the line input on one of the channels of the mixer, and then run the mixer's stereo output to the firewire interface, you may very well end up adding noise to the signal path, but if everything were set preoperly, the mixer's preamps wouldn't effect you, since you should be plugging a preamp's output into the line input (which doesn't hae the same preamp circuitry that XLR inputs have).
On the other hand, had you run the XLR output of the UA (if it has one), to a mic input on your mixer or interface, it might have a preamp that is always active, and thus it would amplify the signal a bit, which is unnecessary in most cases, since you could just turn the gain up on the UA if you needed an extra boost (unless you've maxed out gain there, but I think that is the exception, not the general case).
As an example, I had a Behringer mixer that has "invisibile mic pre's", so when I ran the line-level output of my ADAT recorders to the XLR inputs on it, the signal was getting boost a bit. When I switched the output to the line-in's, the signal levelled out.
Any more questions, feel free to ask!
