  Non-zero neutrino mass would affect the evolution of the Universe in
observable ways, and a strong constraint on the mass can be achieved using
combinations of cosmological data sets. We focus on the power spectrum of
cosmic microwave background (CMB) anisotropies, the Hubble constant H_0, and
the length scale for baryon acoustic oscillations (BAO) to investigate the
constraint on the neutrino mass, m_nu. We analyze data from multiple existing
CMB studies (WMAP5, ACBAR, CBI, BOOMERANG, and QUAD), recent measurement of H_0
(SHOES), with about two times lower uncertainty (5%) than previous estimates,
and recent treatments of BAO from the Sloan Digital Sky Survey (SDSS). We
obtained an upper limit of m_nu < 0.2eV (95% C.L.), for a flat LambdaCDM model.
This is a 40% reduction in the limit derived from previous H_0 estimates and
one-third lower than can be achieved with extant CMB and BAO data. We also
analyze the impact of smaller uncertainty on measurements of H_0 as may be
anticipated in the near term, in combination with CMB data from the Planck
mission, and BAO data from the SDSS/BOSS program. We demonstrate the
possibility of a 5 sigma detection for a fiducial neutrino mass of 0.1eV or a
95% upper limit of 0.04eV for a fiducial of m_nu = 0eV. These constraints are
about 50% better than those achieved without external constraint. We further
investigate the impact on modeling where the dark-energy equation of state is
constant but not necessarily -1, or where a non-flat universe is allowed. In
these cases, the next-generation accuracies of Planck, BOSS, and 1% measurement
of H_0 would all be required to obtain the limit m_nu < 0.05 - 0.06eV (95%
C.L.) for the fiducial of m_nu = 0eV. The independence of systematics argues
for pursuit of both BAO and H_0 measurements.
