Mithril can only be found in Moria, per Gandalf's words in Fellowship of the Ring:
The wealth of Moria was not in gold and jewels, the toys of the Dwarves; nor in iron, their servant. Such things they found here, it is true, especially iron; but they did not need to delve for them: all things that they desired they could obtain in traffic. For here alone in the world was found Moria-silver, or true-silver as some have called it: mithril is the Elvish name.
That doesn't rule out the possibility that it was also to be found in Aman (which was removed from the circles of the world at the time Gandalf spoke); see especially Bilbo's song in Rivendell (which admittedly may have been embellished by Bilbo himself):
A ship then new they built for him
of mithril and of elven-glass
with shining prow; no shaven oar
nor sail she bore on silver mast
This notably happened after Earendil's errand on behalf of the Two Kindreds, so it was a Valinorean event.
Update: 9th March 2015
Footnote 31 to The Disaster of the Gladden Fields in Unfinished Tales confirms that mithril was also to be found in Númenor:
For that metal was found in Númenor. [Author's note.]
Númenor, however, was not in Middle-earth, and Gandalf's words may still be seen to be true ("from a certain point of view") if we assume that he was referring to a time after the Downfall.
What's especially interesting here is that we don't actually know what mithril is at the time of Bilbo's song.