We present optical and near-infrared photometry, as well as ground-based optical spectra and Hubble Space Telescope ultraviolet spectra, of the Type Ia supernova (SN) 2001ay. At maximum light the Si II and Mg II lines indicated expansion velocities of 14,000 km s-(1), while Si III and S II showed velocities of 9000 km s(-1). There is also evidence for some unburned carbon at 12,000 km s(-1). SN 2001ay exhibited a decline-rate parameter of Delta m(15)(B) = 0.68 +/- 0.05 mag; this and the B-band photometry at t greater than or similar to + 25 day past maximum make it the most slowly declining Type Ia SN yet discovered. Three of the four super-Chandrasekhar-mass candidates have decline rates almost as slow as this. After correction for Galactic and host-galaxy extinction, SN 2001ay had M-B = -19.19 and M-V = -19.17 mag at maximum light; thus, it was not overluminous in optical bands. In near-infrared bands it was overluminous only at the 2 sigma level at most. For a rise time of 18 days (explosion to bolometric maximum) the implied Ni-56 yield was (0.58 +/- 0.15)/alpha M-circle dot, with alpha = L-max/E-Ni probably in the range 1.0-1.2. The Ni-56 yield is comparable to that of many Type Ia SNe. The "normal" Ni-56 yield and the typical peak optical brightness suggest that the very broad optical light curve is explained by the trapping of gamma rays in the inner regions.