Observation of the diphoton decay mode of the recently discovered Higgs boson and measurement of some of its properties are reported. The analysis uses the entire dataset collected by the CMS experiment in proton-proton collisions during the 2011 and 2012 LHC running periods. The data samples correspond to integrated luminosities of 5.1 inverse femtobarns at sqrt(s) = 7 TeV and 19.7 inverse femtobarns at 8 TeV. A clear signal is observed in the diphoton channel at a mass close to 125 GeV with a local significance of 5.7 sigma, where a significance of 5.2 sigma is expected for the standard model Higgs boson. The mass is measured to be 124.70 +/- 0.34 GeV = 124.70 +/- 0.31 (stat) +/- 0.15 (syst) GeV, and the best-fit signal strength relative to the standard model prediction is 1.14 +0.26/-0.23 = 1.14 +/- 0.21 (stat) +0.09/-0.05 (syst) +0.13/-0.09 (theo). Additional measurements include the signal strength modifiers associated with different production mechanisms, and hypothesis tests between spin-0 and spin-2 models.