PG 1407+265, discovered in the Palomar-Green Survey (Schmidt & Green 1983), was identified as a z similar to 1 radio-quiet quasar on the basis of a single weak line. Further observations over a wide wavelength range confirm the identification but reveal the object to have unusual emission-line properties. Broad H alpha is the only strong emission line with H beta and Lyman-alpha almost undetectably weak. The emission lines show a range in redshift of over 10,000 km s(-1), systematically decreasing with ionization potential or, almost equivalently, increasing to longer wavelengths. A value of z = 0.94 +/- 0.02 is a reasonable statement of our knowledge of the quasar's redshift. However, the object's continuum properties are those of a normal radio-quiet quasar. We discuss a number of possible models for the object, but its nature remains puzzling.