We report on the first measurement of the double-spin asymmetry, A(LL), of electrons from the decays of hadrons containing heavy flavor in longitudinally polarized p + p collisions at root s = 200 GeV for p(T) = 0.5 to 3.0 GeV/c. The asymmetry was measured at midrapidity (vertical bar eta vertical bar < 0.35) with the PHENIX detector at the Relativistic Heavy Ion Collider. The measured asymmetries are consistent with zero within the statistical errors. We obtained a constraint for the polarized gluon distribution in the proton of vertical bar Delta g/g(log(10)(x) = -1.6(-0.4)(+0.5), mu = m(T)(c)vertical bar(2) < 0.030 (1 sigma) based on a leading-order perturbative quantum chromodynamics model, using the measured asymmetry.