The authors study the convergence rate of complete moment convergence for weighted sums of weakly dependent random variables without assumptions of identical distribution. Under the moment condition of $E{\left|X\right|}^{\alpha }/{\left(\log \left(1+\left|X\right|\right)\right)}^{\alpha /\gamma -1}<\mathrm{\infty }$ for $0<\gamma <\alpha$ with $1<\alpha \le 2$, we establish the complete $\alpha$-th moment convergence theorem for weighted sums of weakly dependent cases, which improves and extends the related known results in the literature.