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The bonding and electrostatic prperties of gold containig molecules are highly influenced by realtivistic effects {\textbf{43}} 4412, 2004.}. Recently Varberg's group reported {\textbf{114}} 4831, 2010.} on the field-free detection and analysis of the $[17.8]0+$$-X1\Sigma +$ band system of AuF at Doppler limited resolution. Here we report on the analysis of the same band recorded field free and in the presence of a static electric field for a cold molecular beam sample. The Stark induced shifts of the R(0)(17756.156 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$), R(1)(17756.640 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$) and P(1)(17755.123 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$)branch features were analyzed to determine the permanent electric dipole moment, $\mu el$, of 2.025 D and 4.218 D for the $[17.8]0+$ and $ X^{1}\Sigma^{+}$ states, respectively. The agreement with the various DFT {\textbf{76}} 032507, 2007.} and wave-function {\textbf{472}} 185, 2009.} based predictions will be discussed.