<p style='text-indent:20px;'>In this article, we propose a new over-penalized weak Galerkin (OPWG) method with a stabilizer for second-order elliptic problems. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements <inline-formula><tex-math id="M1">\begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{d}) $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M2">\begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k-1}, [\mathbb{P}_{k-1}]^{d}) $\end{document}</tex-math></inline-formula>, with dimensions of space <inline-formula><tex-math id="M3">\begin{document}$ d = 2, \; 3 $\end{document}</tex-math></inline-formula>. The method is absolutely stable with a constant penalty parameter, which is independent of mesh size and shape-regularity. We prove that for quasi-uniform triangulations, condition numbers of the stiffness matrices arising from the OPWG method are <inline-formula><tex-math id="M4">\begin{document}$ O(h^{-\beta_{0}(d-1)-1}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \beta_{0} $\end{document}</tex-math></inline-formula> being the penalty exponent. Therefore we introduce a new <i>mini-block diagonal</i> preconditioner, which is proven to be theoretically and numerically effective in reducing the condition numbers of stiffness matrices to the magnitude of <inline-formula><tex-math id="M6">\begin{document}$ O(h^{-2}) $\end{document}</tex-math></inline-formula>. Optimal error estimates in a discrete <inline-formula><tex-math id="M7">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-norm and <inline-formula><tex-math id="M8">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm are established, from which the optimal penalty exponent can be easily chosen. Several numerical examples are presented to demonstrate flexibility, effectiveness and reliability of the new method.