Does the fine-structure constant vary with cosmological epoch?

We use the strong nebular emission lines of O III, lambdalambda5007 and 4959, to set a robust upper limit on the time dependence of the fine-structure constant. We find \alpha(-1) dalpha(t)/dt\ < 2 x 10(-13) yr(-1), corresponding to Delta alpha/alpha(0) = (0.7 +/- 1.4) x 10(-4) for quasars with 0.16 < z < 0.80 obtained from the SDSS Early Data Release. Using a blind analysis, we show that the upper limit given here is invariant with respect to 17 different ways of selecting the sample and analyzing the data. As a by-product, we show that the ratio of transition probabilities corresponding to the lambda lambda 5007 and 4959 lines is 2.99 +/- 0.02, in good agreement with ( but more accurate than) theoretical estimates. We compare and contrast the O III emission line method used here with the many-multiplet method that was used recently to suggest evidence for a time-dependent alpha. In an appendix we analyze the quasars from the recently available SDSS Data Release One sample and find Delta alpha/alpha(0) = (1.2 +/- 0.7) x 10(-4).