Question: Emigrant = someone who is leaving their country.

??? = the country from which the emigrant is departing.

I want to say Émigré country but I don't know if that makes sense.

Maybe country of emigration? But that's too wordy.

Is there *one word*?

Answer: 
This depends on your audience. A speech would be different than a novel.

In nonfiction writing or speech, "country of origin" is the most respectful and politically-sensitive way of phrasing this. 

"Home country" doesn't work if the person has neither a home nor feels at home there.

"Native country" also doesn't work because it may not be where someone is native from. Consider a refugee of palestine who has emigrated from Egypt. While their native country may be Palestine/Israel, if they sought temporary political asylum in Egypt, and then emigrated to England, their country of origin would be Egypt.

In fiction and informal writing, "homeland."
In most cases, it would be **homeland** or **motherland**.

But "**[old country](http://www.merriam-webster.com/dictionary/old%20country)**" is used also

> 
> an emigrant's country of origin
> 
> 
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Additionally, **"source country"** is used in immigration related or technical sources