We present variants of the singlet-Majoron model with the seesaw mechanism effective at the weak scale that accommodate the recently reported 17-keV neutrino (nu-17) naturally. First, we show that within the minimal model, by assuming an unbroken global l(e) - l(mu) + l(tau) symmetry (l(i)=ith lepton number), nu-17 can be identified as a Dirac particle composed of nu(tau) and nu(mu)BAR. It is then shown that, with the same spectrum, if the l(e) - l(mu) + l(tau), symmetry is broken spontaneously below the weak scale, nu(e)-nu(s) (nu(s) stands for sterile neutrino) oscillations can account for the solar-neutrino deficit via the Mikheyev-Smirnov-Wolfenstein (MSW) mechanism. We also derive a recently proposed mass matrix for the 17-keV neutrino, which features nu(e)-nu(s) MSW oscillations, within the context of the seesaw model. All known constraints from cosmology and astrophysics, including the supernova constraint on the mass of nu-17 are satisfied in these variants.