Is it possible to make deuterated organic compounds?
Deuterium occurs at around the 0.02% level in nature (natural abundance level).  So all compounds that contain hydrogen will already contain a low level concentration of the deuterated isomer(s).  So in fact, deuterated compounds do exist in nature. 

Despite this low level of natural deuterium incorporation, the deuterium can still be detected by various spectroscopic techniques (mass spectrometry and nmr for example).

Further, organic compounds can be synthesized with deuterium in specific locations. These deuterated compounds are still considered organic compounds. Since deuterium is a stable nucleus (e.g. does not undergo radioactive decay), the compounds containing deuterium are also stable.  Because the $\ce{C-D}$ bond is a bit shorter and therefore a bit stronger than the corresponding $\ce{C-H}$ bond, there is an isotope effect in reactions where these bonds are broken. Because the $\ce{C-H}$ bond is weaker, it will break up to 7 times faster than the corresponding $\ce{C-D}$ bond. This can have a significant effect on biological processes.

$\ce{D2O}$ and $\ce{D2}$ gas are relatively common and inexpensive materials that can be used to introduce deuterium into other molecules. Here are 2 examples involving the use of these materials for deuterium incorporation.  In the first example $\ce{D2O}$ is used as the source of deuterium, in the second example deuterium gas is used to introduce the deuterium.  If you google "organic preparation of deuterated compounds" you'll find many more examples.



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Specifically deuterated compounds are widely used in many areas of research and industry.