  We show that at least 19/27 of the zeros of the Riemann zeta-function are
simple, assuming the Riemann Hypothesis (RH). This was previously established
by Conrey, Ghosh and Gonek [Proc. London Math. Soc. 76 (1998), 497--522] under
the additional assumption of the Generalised Lindel\"of Hypothesis (GLH). We
are able to remove this hypothesis by careful use of the generalised Vaughan
identity.
