One option would be to use the titlesec
package to locally redefine the section formatting:
\documentclass{article}
\usepackage{titlesec}
\begin{document}
\section{Test Section One}
\begingroup
\titleformat*{\section}{\normalfont}
\begin{thebibliography}{depth}
\bibitem{a} Test
\end{thebibliography}
\endgroup
\section{Test Section Two}
\end{document}

Another option is to patch the \thebibliography
command to replace the default \section*{\refname}
with \refname
; this can be easily done with the help of the etoolbox
package:
\documentclass{article}
\usepackage{etoolbox}
\patchcmd{\thebibliography}{\section*{\refname}}{\refname}{}{}
\begin{document}
\section{Test Section One}
\begin{thebibliography}{depth}
\bibitem{a} Test
\end{thebibliography}
\section{Test Section Two}
\end{document}
Without any packages, the necessary redefinition would be:
\documentclass{article}
\makeatletter
\renewenvironment{thebibliography}[1]
{\refname%
\@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%
\list{\@biblabel{\@arabic\c@enumiv}}%
{\settowidth\labelwidth{\@biblabel{#1}}%
\leftmargin\labelwidth
\advance\leftmargin\labelsep
\@openbib@code
\usecounter{enumiv}%
\let\p@enumiv\@empty
\renewcommand\theenumiv{\@arabic\c@enumiv}}%
\sloppy
\clubpenalty4000
\@clubpenalty \clubpenalty
\widowpenalty4000%
\sfcode`\.\@m}
{\def\@noitemerr
{\@latex@warning{Empty `thebibliography' environment}}%
\endlist}
\makeatother
\begin{document}
\section{Test Section One}
\begin{thebibliography}{depth}
\bibitem{a} Test
\end{thebibliography}
\section{Test Section Two}
\end{document}
The last two examples produce the same output as the first one, so I didn't upload the repeated images.