Let
$k$
be a cyclic extension of odd prime degree
$p$
of the field of rational numbers. If
$t$
denotes the number of primes that ramify in
$k$
, it is known that the Hilbert
$p$
-class field tower of
$k$
is infinite if
$t\,>\,3\,+\,2\sqrt{p}$
. For each
$t\,>\,2\,+\,\sqrt{p}$
, this paper shows that a positive proportion of such fields
$k$
have infinite Hilbert
$p$
-class field towers.