The Q(weak) experiment has measured the parity-violating asymmetry in (e) over right arrowp elastic scattering at Q(2) = 0.025 (GeV/c)(2), employing 145 mu A of 89% longitudinally polarized electrons on a 34.4 cm long liquid hydrogen target at Jefferson Lab. The results of the experiment's commissioning run, constituting approximately 4% of the data collected in the experiment, are reported here. From these initial results, the measured asymmetry is A(ep) = -279 +/- 35 (stat) +/- 31 (syst) ppb, which is the smallest and most precise asymmetry ever measured in (e) over right arrowp scattering. The small Q(2) of this experiment has made possible the first determination of the weak charge of the proton Q(W)(p) by incorporating earlier parity-violating electron scattering (PVES) data at higher Q(2) to constrain hadronic corrections. The value of Q(W)(p) obtained in this way is Q(W)(p) (PVES) = 0.064 +/- 0.012, which is in good agreement with the standard model prediction of Q(W)(p) (SM) = 0.0710 +/- 0.0007. When this result is further combined with the Cs atomic parity violation (APV) measurement, significant constraints on the weak charges of the up and down quarks can also be extracted. That PVES + APV analysis reveals the neutron's weak charge to be Q(W)(n) (PVES + APV) = -0.975 +/- 0.010.