Assuming that you're using a book or report document class, add the etoolbox package to your document preamble
\usepackage{etoolbox}% http://ctan.org/pkg/etoolbox
and then use
\appendix
\chapter{List of Publications}
\patchcmd{\thebibliography}{\chapter*{\bibname}}{}{}{}
\begin{thebibliography}{1}
If you're not using thebibliography anywhere else in the document, you could move the \patchcmd{<macro>}{<search>}{<replace>}{<success>}{<failure>} construct to the document preamble. This way you keep document elements and formatting clean.
What is behind this correction? You'll notice that both book and report define the bibliography as a \chapter*, which opens up a new page - exactly what you don't want:
\newenvironment{thebibliography}[1]
{\chapter*{\bibname}%
\@mkboth{\MakeUppercase\bibname}{\MakeUppercase\bibname}%
%...
\patchcmd removes this from the environment. It also typesets the headers with \bibname. Not sure whether this is something you're after. Regardless, it can either be removed (with little hassle, similar to the above correction), or left in.
Of course, if this is the very last entry to your document, and you won't be using a \chapter* anymore, a simple redefinition should also work:
\def\@schapter#1{}
\@schapter is the starred version of \chapter, as defined in latex.ltx. If you have some other \chapter* commands following thebibliography (which is perfectly fine), you could also perform this modification locally by grouping it.