  We show the presence of a first-order phase transition for a ferromagnetic
Ising model on $\mathbb{Z}^2$ with a periodical external magnetic field. The
external field takes two values $h$ and $-h$, where $h>0$. The sites associated
with positive and negative values of external field form a cell-board
configuration with rectangular cells of sides $L_1\times L_2$ sites, such that
the total value of the external field is zero. The phase transition holds if
$h<\frac{2J}{L_1}+ \frac{2J}{L_2}$, where $J$ is an interaction constant. We
prove a first-order phase transition using the reflection positivity (RP)
method. We apply a key inequality which is usually referred to as the
chessboard estimate.
