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: 
[Native country](http://www.collinsdictionary.com/dictionary/english/native-country)  

*the country someone is born in or native to*

> 
> Born Carmela Drelano, in Spain, it was very many years since she had
>  lived in her native country.
> 
> 
> 

[Native land](http://dictionary.reference.com/browse/native%20land) 

> 
> When I think of my own native land, In a moment I seem to be there; 
>  But alas! recollection at hand Soon hurries me back to despair
> 
> 
> 

[Native soil](http://en.wiktionary.org/wiki/native_soil)  

*The country or geographical region where one was born or which one considers to be one's true homeland*

> 
> Nawaz Sharif, two-time Prime Minister of Pakistan, had planned a
>  triumphant return to his native soil nearly seven years after choosing
>  exile.
> 
> 
> 

If the OP wishes a similar one word expression still connected to *native*, then I suggest  

[birthplace](http://dictionary.reference.com/browse/birthplace?&o=100074&s=t)

> 
> * He knows what spot this is: the birthplace of their country.
> * At the age of 27, Arriaga **emigrated from his birthplace**, the port of
>  Callao, Peru, to Canada. [source](http://marcosarriaga.com/promised-land/)
> * [Pulitzer](http://www.stlmediahistory.com/index.php/Print/PrintHOFDetail/pulitzer-joseph) emigrated from **his birthplace in Hungary** to New York in 1864 when he was 17.
> 
> 
>
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