  We continue the study of renewal contact processes initiated in a companion
paper, where we showed that if the tail of the interarrival distribution $\mu$
is heavier than $t^{-\alpha}$ for some $\alpha <1$ (plus auxiliary regularity
conditions) then the critical value vanishes. In this paper we show that if
$\mu$ has decreasing hazard rate and tail bounded by $t^{-\alpha}$ with $\alpha
>1$, then the critical value is positive in the one-dimensional case. A more
robust and much simpler argument shows that the critical value is positive in
any dimension whenever the interarrival distribution has a finite second
moment.
