We observe a critical double point (CDP) in a polystyrene (molecular weight = 13 000) plus acetone mixture when a small amount of water is added to it instead of varying the molecular weight or the pressure. A very close access to CDP has been achieved, as exemplified by a sample with a miscibility gap, Delta T-E (the difference between the upper and lower extremum temperatures, T-EU and T-EL, respectively) as small as 194 mK. The critical exponent (beta) for the phase diagram exhibits a twofold increase in its 3-D Ising value for a sample with Delta T-E=194 mK, when the usual field variable (the reduced temperature, t) is used. The universal value of beta(=0.325) is retrieved for any Delta T-E with the aid of the field variable t(UL)[=(T-EU-T)(T-T-EL)/TEUTEL]. Our evidence shows that, at least for the order parameter, Delta T-E is preferable to a Delta T-c (the difference between the upper and lower critical temperatures, T-U and T-L, respectively) as a measure of the distance from CDP in this system. The critical line and the locus of the extremum temperatures are similar and are nearly parabolic in shape. Most of our observations can be understood in terms of the geometric picture of phase transitions as well as the Landau-Ginzburg theory, as applied to reentrant phase transitions. A microscopic understanding of the role of water in reducing the miscibility gap (Delta T-E) in these complex systems remains to be achieved.