I might have misunderstood the question... but I think you want to wrap the quotation within text. It can be achieved using wrapfig package.
\documentclass{article}
\usepackage{wrapfig}% http://ctan.org/pkg/wrapfig
\usepackage{lipsum}% http://ctan.org/pkg/lipsum
\begin{document}
\begin{wrapfigure}[11]{l}{.5\textwidth}
\fbox{\begin{minipage}{\dimexpr\linewidth-2\fboxrule-2\fboxsep}
%\centering
``Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Proin ac tempus dui. Etiam non mattis neque. Maecenas tempor,
massa a luctus sollicitudin, elit massa auctor velit, eu mattis
lacus arcu nec est.
Sed et lectus non tellus posuere consequat.
Donec congue lacinia urna a iaculis. Cras id nisl sed sapien
euismod rhoncus.''
\end{minipage}}
\end{wrapfigure}
\lipsum[1]
\end{document}

If you want the box to span many pages, then the mdframed option suggested in the other answer should be followed.
figureis doing this. – Stephan Lehmke Apr 18 '12 at 8:50figureis what I need, but I am not sure iffigureis the best choice for text. – All Apr 18 '12 at 10:51figureenvironment is not wrapped in text at all. Are you talking about a specific package? – Stephan Lehmke Apr 18 '12 at 11:00wrapfig. In the original question, I referred to the possibility to wrapfigureby corresponding packages. And I am asking for the best package to wrap a box of text. – All Apr 18 '12 at 11:10