I was in Alliance with one empire, but my enemy broke this alliance through fake communication. Now I in the war with my ex-ally. Is it possible to return my ex-ally in the alliance? Do you have such experience?
Is this an AI or a human empire? If AI, you have to go for so many turns without destroying anything of theirs (20 turns or so for many AIs) before they will accept a treaty with you again.
Once I'm at war with an AI, I have never been able to make peace with them. I didn't know that you could recover by not destroying their stuff. ...OK, I'll try not to destroy anything, but only if they don't attack me.
Several things are documented in the individual AI_Anger.txt files. I haven't read them extensively -- I like a little wonder. But here's something: The AI get more angry when you destroy it's ships in combat, and also when it destroys yours (it's winning after all, might as well press the advantage). So if you're playing tactical combat, just don't blow up the ships. If you're using strategic combat, use shield depleters and engine destroyers. As the AI starts losing income, it soon sues for peace.