See the Transparency and Blending manual page.
David
I’ve tried all the modes on that page, and none of them work.
I feel like we’re getting close though. 
Well, if you don’t expect transparency at all, you could replace the alpha-mode with “none”. Do your textures have an alpha channel? You should consider removing the alpha channel if you don’t want to use it for transparency.
David
Weird thing is, it did have an alpha channel, but removing it didn’t change anything. 
And changing the alpha mode to ‘none’ gives me “unknown alpha mode none”.
Ok, just had a quick look, two things:
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Your trick of setting ‘TexFace’ and ‘A’ works, but the provided mesh has 5 materials applied and you only set this for 4 of them, hence why part of your mesh continues to appear transparent.
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Your current trick is really just overriding the fact all your materials have a separate ‘Opacity’ texture applied - if you want to lose the transparency just untick/delete the opacity channels and the problem will go away - that is the correct approach.
Removing the opacity channel partly works, thanks. Now it’s not translucent, but I can still see things behind it. Pic: http://i54.tinypic.com/2wcntsg.jpg. We’re almost done, I can feel it.
EDIT: Wait, I set the alpha mode to ‘dual’ and it’s perfect now! 
I feel like an idiot now though, one, for not noticing the opacity thing, and two, for not trying different alpha modes before posting this. 
Still, all of you were very helpful. I probably would have quit by now if it weren’t for you guys. 
Sorry, though, you should set the alpha mode to “off” if you really don’t want transparency at all (not “none”, my mistake). “dual” is a fairly expensive two-pass rendering mode that’s a great big waste if you’re not actually using transparency.
David
Okay, ‘off’ works too.