We have performed laser spectroscpy of metastable antiprotonic helium atoms (or "atomcules") ((p) over bar He+) and have observed a density dependence of the resonance vacuum wavelengths for the known transitions (n,l)=(39,35)-->(38,34) and (37,34)-->(36,33). They showed linear red-shifts of 0.61+/-0.01 GHz and 0.22+/-0.02 GHz per 1 g/l, respectively. With the shift parameters above, the transition vacuum wavelengths were extrapolated to zero-density limits, yielding lambda(o) = 597.2570+/-0.0003 nm and lambda(o) = 470.7220 +/- 0.0006 nm, respectively. These values were compared with the result of recent theoretical calculations on the energy of the Coulombic three-body system, including relativistic corrections and the Lamb shift. The agreements between our experimental values and the calculations have become as good as 2x10(-6). This sets a severe constraint on the antiproton charge (-Q((p) over bar)) and mass (M-(p) over bar) with >M-p - M((p) over bar)nder a more precisely known constraint on the charge-to-mass ratio. Thus we have opened a new possibility of measuring fundamental constants of the antiproton.