  We give direct experimental evidence for the observation of the full
transverse self-modulation of a relativistic proton bunch propagating through a
dense plasma. The bunch exits the plasma with a density modulation resulting
from radial wakefield effects with a period reciprocal to the plasma frequency.
We show that the modulation is seeded by using an intense laser pulse
co-propagating with the proton bunch which creates a relativistic ionization
front within the bunch. We show by varying the plasma density over one order of
magnitude that the modulation period scales with the expected dependence on the
plasma density.
