The weight of one cubic centimeter of water is in fact one gram. By definition.
Although 1 cm^3 of water is still pretty damn close to 1g.
[1] https://en.wikipedia.org/wiki/Grave_%28unit%29
[2] https://www.youtube.com/watch?v=ZMByI4s-D-Y (around 1:50, and actually the whole video is excellent)