**There are multiple, historical examples of artillery/mortars firing stone balls.**  Wikipedia has an entry (with photos, including some stone balls) showing only the biggest ones.  So there's enough of a historical record that these weren't total death traps for the crews working them.  I suspect there were many more of smaller caliber: 
[https://en.wikipedia.org/wiki/List_of_the_largest_cannon_by_caliber#Stone_balls][1]

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**If you are willing to accept the speculative, a stone cannon is possible -- without being carved from an existing block of stone. Instead, it could be made from two kinds of molten stone.**  Probably re-melted stone, rather than lava, since we need two different types, one to produce strong, elastic fibers and a second (with a lower melting point) to form the matrix that the tensile fibers are embedded into, as follows.  

**Basalt fibers are a real, usefully strong, commercially available thing.** See 
[en.wikipedia.org/wiki/Basalt_fiber][2]  
and  
[basalt-fiber.com][3]   
So, one could easily imagine using basalt fibers, in some suitable matrix, to make a cannon barrel out of stone.  We first produce basalt fibers, which have yield stresses and strains very similar to glass fibers.  
Some of the mechanical and thermal properties are summarized here: 
[http://smarter-building-systems.com/smarter-building-basalt-faqs/][4]

I posit that an all-stone-derived fiber-reinforced ceramic material would give us much of the strength of the fibers, so long as we: 

- make the fibers dominate the volume. Namely get as close to (pi/4 as optimal long cylinder packing, aligned case of course) as we can.

- embed the fibers in another melted mineral, so long as it has a lower melting temperature than the basalt and is chemically compatible, at least short term.  From 
[http://www.minsocam.org/msa/collectors_corner/arc/tempmagmas.htm][5]  
it's evident that we could use a rhyolitic mineral for the matrix material, for the lower melting temperature.  There are alternatives, and 
Cryolite (Na3AlF6) has a fairly low melting point: 
[http://www.gly.uga.edu/railsback/Fundamentals/HardnessMeltingPlot06.pdf][6]

- The fiber is wound on a removable (or sacrificial) mandrel.  The orientation of the fibers can be changed (even on a layer by layer basis) to optimize the strength and shock/fracture toughness.  

- The wound fiber is placed in a mold (ideally a sealed, evacuated mold), preferably pre-heated, and then the molten-rock matrix is poured in.  

- After the casting cools, the bore can be ground out to the desired diameter.  With sufficiently localized heating (and a sufficiently large calliber to permit access), such a gun barrel could be repaired and maintained.  

By judicious casting, a rifled barrel is not entirely beyond possibility.  
This would only make sense if virtually no usable metal were available -- but such a world might exist.

  [1]: https://en.wikipedia.org/wiki/List_of_the_largest_cannon_by_caliber#Stone_balls
  [2]: http://en.wikipedia.org/wiki/Basalt_fiber
  [3]: http://basalt-fiber.com
  [4]: http://smarter-building-systems.com/smarter-building-basalt-faqs/
  [5]: http://www.minsocam.org/msa/collectors_corner/arc/tempmagmas.htm
  [6]: http://www.gly.uga.edu/railsback/Fundamentals/HardnessMeltingPlot06.pdf