We report new limits on the Gunn-Peterson effect at a redshift near 5 using spectroscopic observations of the z = 5 Sloan Digital Sky Survey quasar, J033829.31+002156.3, made with the Low Resolution Imaging Spectrometer (LRIS) and the High-Resolution Echelle Spectrometer (HIRES) on the Keck telescopes. Lower resolution spectrophotometrically calibrated observations made with LRIS over the wavelength region 4500-9600 Angstrom were used to obtain a continuum shape and to flux-calibrate much higher resolution (R = 36,000) observations made with HIRES. The LRIS data show an Oke D-A index of 0.75. Portions of the HIRES spectrum return to near the extrapolated continuum level. Including both statistical and systematic errors, we place an upper limit of tau = 0.1 on the regions of minimum opacity. We argue that, even if this opacity arises in underdense regions of the intergalactic gas, we require a high value of the metagalactic ionizing flux at these redshifts (J(y)>> 4 x 10(-23) ergs cm(-2) s(-1) Hz(-1) sr(-1) at z similar to 4.72) to produce a solution which is consistent with even minimum nucleosynthesis estimates of the baryon density. We also report the presence of an Mg II absorption system of extremely high equivalent width [W-lambda,W-rest (2796)= 1.73 Angstrom] at z = 2.304.