  We consider the ill-posedness issue for the cubic nonlinear heat equation and
prove norm inflation with infinite loss of regularity in the H\"older-Besov
space $\mathcal C^s = B^{s}_{\infty, \infty}$ for $ s \le -\frac 23$. In
particular, our result includes the subcritical range $-1< s \le -\frac 23$,
which is above the scaling critical regularity $s = -1$ with respect to the
H\"older-Besov scale. In view of the well-posedness result in $\mathcal C^s$,
$s > -\frac 23$, our ill-posedness result is sharp.
