If you do not mind using thebibliography
instead of \bibliography
you can also use tcolorbox
\documentclass{beamer}
\usetheme{Warsaw}
\usepackage{tcolorbox}
\tcbuselibrary{skins,breakable}
\tcbset{breakable}
\setbeamertemplate{bibliography item}{}
\begin{document}
\begin{frame}[allowframebreaks]{References}
\begin{thebibliography}{}
\begin{tcolorbox}[beamer, skin=beamerfirst, colback=blue!5!white, colframe=blue!50!black, title=Bibliography]
\bibitem[{Alpher} et~al.(1948) {Alpher}, {Bethe}, \& {Gamow}]{paper}
Alpher, R. A., Bethe, H. \& Gamow, G.
\newblock{The Origin of Chemical Elements}.
\newblock{\emph{Phys. Rev.}, 73:\penalty0 803--804, Apr. 1948.}
\newblock{http://dx.doi.org/10.1103/PhysRev.73.803}
\bibitem[{Alpher} et~al.(1948) {Alpher}, {Bethe}, \& {Gamow}]{paper2}
Alpher, R. A., Bethe, H. \& Gamow, G.
\newblock{The Origin of Chemical Elements}.
\newblock{\emph{Phys. Rev.}, 73:\penalty0 803--804, Apr. 1948.}
\newblock{http://dx.doi.org/10.1103/PhysRev.73.803}
\bibitem[{Alpher} et~al.(1948) {Alpher}, {Bethe}, \& {Gamow}]{paper3}
Alpher, R. A., Bethe, H. \& Gamow, G.
\newblock{The Origin of Chemical Elements}.
\newblock{\emph{Phys. Rev.}, 73:\penalty0 803--804, Apr. 1948.}
\newblock{http://dx.doi.org/10.1103/PhysRev.73.803}
\newpage
\bibitem[{Alpher} et~al.(1948) {Alpher}, {Bethe}, \& {Gamow}]{paper4}
Alpher, R. A., Bethe, H. \& Gamow, G.
\newblock{The Origin of Chemical Elements}.
\newblock{\emph{Phys. Rev.}, 73:\penalty0 803--804, Apr. 1948.}
\newblock{http://dx.doi.org/10.1103/PhysRev.73.803}
\end{tcolorbox}
\end{thebibliography}
\vfill
\end{frame}
\end{document}
