This is the second of two papers reporting results from a program to determine the Hubble constant to ~5% precision from a refurbished distance ladder based on extensive use of differential measurements. Here we report observations of 240 Cepheid variables obtained with the Near-Infrared Camera and Multi-Object Spectrometer (NICMOS) Camera 2 through the F160W filter on the Hubble Space Telescope (HST). The Cepheids are distributed across six recent hosts of Type Ia supernovae (SNe Ia) and the "maser galaxy" NGC 4258, allowing us to directly calibrate the peak luminosities of the SNe Ia from the precise, geometric distance measurements provided by the masers. New features of our measurement include the use of the same instrument for all Cepheid measurements across the distance ladder and homogeneity of the Cepheid periods and metallicities, thus necessitating only a differential measurement of Cepheid fluxes and reducing the largest systematic uncertainties in the determination of the fiducial SN Ia luminosity. In addition, the NICMOS measurements reduce the effects of differential extinction in the host galaxies by a factor of ~5 over past optical data. Combined with a greatly expanded set of 240 SNe Ia at z < 0.1 which define their magnitude-redshift relation, we find H ₀ = 74.2 ± 3.6 km s^–1 Mpc^–1, a 4.8% uncertainty including both statistical and systematic errors. To independently test the maser calibration, we use 10 individual parallax measurements of Galactic Cepheids obtained with the HST fine guidance sensor and find similar results. We show that the factor of 2.2 improvement in the precision of H ₀ is a significant aid to the determination of the equation-of-state parameter of dark energy, w = P/(ρc ²). Combined with the Wilkinson Microwave Anisotropy Probe five-year measurement of Ω _M h ², we find w = –1.12 ± 0.12 independent of any information from high-redshift SNe Ia or baryon acoustic oscillations (BAO). This result is also consistent with analyses based on the combination of high-redshift SNe Ia and BAO. The constraints on w(z) now including high-redshift SNe Ia and BAO are consistent with a cosmological constant and are improved by a factor of 3 due to the refinement in H ₀ alone. We show that future improvements in the measurement of H ₀ are likely and should further contribute to multi-technique studies of dark energy.