  We consider the real part $\Re(\zeta(s))$ of the Riemann zeta-function
$\zeta(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately
the constant $\sigma_0 = 1.19\ldots$ which is defined to be the supremum of
$\sigma$ such that $\Re(\zeta(\sigma+it))$ can be negative (or zero) for some
real $t$. We also consider intervals where $\Re(\zeta(1+it)) \le 0$ and show
that they are rare. The first occurs for $t$ approximately 682112.9, and has
length about 0.05. We list the first fifty such intervals.
