  The paper is devoted to the relationship between almost limited operators and
weakly compacts operators. We show that if $F$ is a $\sigma $-Dedekind complete
Banach lattice then, every almost limited operator $T:E\rightarrow F $ is
weakly compact if and only if $E$ is reflexive or the norm of $F$ is order
continuous. Also, we show that if $E$ is a $\sigma $-Dedekind complete Banach
lattice then the square of every positive almost limited operator $
T:E\rightarrow E$ is weakly compact if and only if the norm of $E$ is order
continuous.
