Using weight functions, the authors establish that the following Hilbert-type multiple integral inequality \[ \int_{\mathbb{R^{n}_{+}}} \int_{\mathbb{R^{n}_{+}}}K\Big(\mid\mid x\mid\mid_{m,\rho},\mid\mid y\mid\mid_{n,\rho}\Big)f(x)g(y)dxdy \le M\mid\mid f\mid\mid_{p,\alpha}\mid\mid g\mid\mid_{q,\beta}\] holds true if and only if \[\frac{\alpha +m}{p}+ \frac{\beta +n}{q}= \lambda + m+ n,\] and the expression of the best possible constant factor in the above inequality is derived and proved. Furthermore, its application in operator theory is discussed.