We present precise phase-connected pulse timing solutions for 16 gamma-ray-selected pulsars recently discovered using the Large Area Telescope (LAT) on the Fermi Gamma-ray Space Telescope plus one very faint radio pulsar (PSR J1124-5916) that is more effectively timed with the LAT. We describe the analysis techniques including a maximum likelihood method for determining pulse times of arrival from unbinned photon data. A major result of this work is improved position determinations, which are crucial for multi-wavelength follow up. For most of the pulsars, we overlay the timing localizations on X-ray images from Swift and describe the status of X-ray counterpart associations. We report glitches measured in PSRs J0007+7303, J1124-5916, and J1813-1246. We analyze a new 20 ks Chandra ACIS observation of PSR J0633+0632 that reveals an arcminute-scale X-ray nebula extending to the south of the pulsar. We were also able to precisely localize the X-ray point source counterpart to the pulsar and find a spectrum that can be described by an absorbed blackbody or neutron star atmosphere with a hard powerlaw component. Another Chandra ACIS image of PSR J1732-3131 reveals a faint X-ray point source at a location consistent with the timing position of the pulsar. Finally, we present a compilation of new and archival searches for radio pulsations from each of the gamma-ray-selected pulsars as well as a new Parkes radio observation of PSR J1124-5916 to establish the gamma-ray to radio phase offset.
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