We answer positively a conjecture of Arratia (1983) that the position of the right-most particle in the simple symmetric exclusion process on Z, when begun from a step profile, obeys the same Gumbel limit law as a system of independent particles with the same initial condition, in the limit as time goes to infinity. More generally, we consider initial profiles from a class of periodic Bernoulli-step initial measures and show that Gumbel limits hold for the properly-scaled leading particle position in the symmetric exclusion process when the single particle jump rate has finite moment generating function. Moreover, to investigate the influence of the mass of particles behind the leading one, we also consider initial profiles consisting of a periodic block of L particles, where L goes to infinity in time. Gumbel limit laws are obtained for the leading particle position under appropriate scaling for all rates at which L diverges in time. In particular, there is a phase transition when L is of order t^1/2(log t)^-1/2, above which the asymptotics match those of the infinite step profile, and below which the displacement is of order (t log L)^1/2. This is joint work with Sunder Sethuraman.