A new six-membered carbonate functionalized with 2-(morpholin-4-yl)ethyl was synthesized and copolymerized with a poly(ethylene oxide)-based cross-linker in a controlled, organocatalyzed ring-opening polymerization process. Well-defined functionalized aliphatic polycarbonate-based hydrogels were obtained with gel fractions reaching 99%. Moreover, the gels that contained 2-(morpholin-4-yl)ethyl groups revealed stimuli-responsive properties towards pH. In the meantime, swelling properties of the hydrogels were found to be reproducible and good stability was observed in the pH range of 4–7.2, while a relatively rapid degradation occurred in more basic solution (pH = 10). ICP MS measurements of aqueous lead(II) nitrate solutions revealed that the functionalized material possessed properties to withdraw and retain lead ions from aqueous solution.