  We present the exact solution of the 1D classical short-range Potts model
with invisible states. Besides the $q$ states of the ordinary Potts model, this
possesses $r$ additional states which contribute to the entropy, but not to the
interaction energy. We determine the partition function, using the
transfer-matrix method, in the general case of two ordering fields: $h_1$
acting on a visible state and $h_2$ on an invisible state. We analyse its zeros
in the complex-temperature plane in the case that $h_1=0$. When ${\rm Im}\,
h_2=0$ and $r\ge 0$, these zeros accumulate along a line that intersects the
real temperature axis at the origin. This corresponds to the usual "phase
transition" in a $1$D system. However, for ${\rm Im}\, h_2\neq 0$ or $r<0$, the
line of zeros intersects the positive part of the real temperature axis, which
signals the existence of a phase transition at non-zero temperature.
