Just take the elements from the title page an arrange them to your needs
\documentclass{beamer}
\usetheme{Warsaw}
\title{Very interesting science}
\author{A.\,Einstein}
\date{31. April 1592}
\institute{My Institut}
\begin{document}
\begin{frame}[plain]
\titlepage
\end{frame}
\begin{frame}[plain]
\vfill
\centering
\begin{beamercolorbox}[sep=8pt,center,colsep=-4bp,rounded=true,shadow=true]{institute}
\usebeamerfont{institute}\insertinstitute
\end{beamercolorbox}
{\usebeamercolor[fg]{titlegraphic}\inserttitlegraphic\par}
\begin{beamercolorbox}[sep=8pt,center,colsep=-4bp,rounded=true,shadow=true]{title}
\usebeamerfont{title}\inserttitle\par%
\ifx\insertsubtitle\@empty%
\else%
\vskip0.25em%
{\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par}%
\fi%
\end{beamercolorbox}%
\vskip1em\par
\begin{beamercolorbox}[sep=8pt,center,colsep=-4bp,rounded=true,shadow=true]{author}
\usebeamerfont{author}\insertauthor
\end{beamercolorbox}
\begin{beamercolorbox}[sep=8pt,center,colsep=-4bp,rounded=true,shadow=true]{date}
\usebeamerfont{date}\insertdate
\end{beamercolorbox}\vskip0.5em
\end{frame}
\end{document}
