How to remove nitrate from water sample?
Precipitation won't work

Nitrate removal is a very tough problem in wastewater treatment.  One of the reasons is that nitrate salts of nearly every common cation are highly water soluble.  Unfortunately, I can't think of a way to precipitate nitrate from your water sample.

How commercial nitrate removal processes work

Nitrate is removed from wastewater through a microbiological process called denitrification.  In this process, organic carbon substrates are used as food by special anaerobic bacteria.  In the absence of oxygen, the bacteria use nitrate as an electron acceptor so they can oxidize their food and obtain energy via respiration.  As an example organic food molecule, consider hexose sugar.  Then the reaction could be written as:

$$\ce{5 C6H12O6 + 24NO3- -&gt; 30CO2 + 24 OH- + 18H2O + 12N2} $$

It will be difficult to purify a few mL of water in this way unless you know exactly the BOD of the water and are an experienced microbiologist with access to cultures of the appropriate bacteria.

Another not very good option

Another possibility would be electrochemical removal of the nitrate.  That is a topic which has been researched but which is not practices commercially to any significant extent.  Here is one paper where you can read about the technique.  I don't think it would be very easy for a home chemist either.

The tried and true

That leaves more tried and true methods.  You've simply got to distill your water!  Boil it and collect and condense the steam and you'll have substantially purified water to behold and enjoy!