  The partition function of the square lattice Ising model on the rectangle
with open boundary conditions in both directions is calculated exactly for
arbitrary system size $L\times M$ and temperature. We start with the dimer
method of Kasteleyn, McCoy & Wu, construct a highly symmetric block transfer
matrix and derive a factorization of the involved determinant, effectively
decomposing the free energy of the system into two parts,
$F(L,M)=F_\mathrm{strip}(L,M)+F_\mathrm{strip}^\mathrm{res}(L,M)$, where the
residual part $F_\mathrm{strip}^\mathrm{res}(L,M)$ contains the nontrivial
finite-$L$ contributions for fixed $M$. It is given by the determinant of a
$\frac{M}{2}\times \frac{M}{2}$ matrix and can be mapped onto an effective spin
model with $M$ Ising spins and long-range interactions. While
$F_\mathrm{strip}^\mathrm{res}(L,M)$ becomes exponentially small for large
$L/M$ or off-critical temperatures, it leads to important finite-size effects
such as the critical Casimir force near criticality. The relations to the
Casimir potential and the Casimir force are discussed.
