First, here is a solution that uses only the xcolor package and the standard verbatim environment. But it is not a new environment, per se, and so does not directly answer the OP's question (for that, see below).
\documentclass{article}
\usepackage{xcolor}
\fboxrule=1pt
\begin{document}
\setbox0=\hbox{\begin{minipage}{3in}
\color{red!90}
\begin{verbatim}
\verbatim <Text>
Here
%$#@&^* \macros
\end{verbatim}
\end{minipage}}
\fcolorbox{cyan}{blue!10}{\box0}
\end{document}

To create an environment as desired by the OP, I adapted color into the syntax employed by the verbatimbox package, and created the cverbbox environment, which stores your verbatim content in a user-named box that is exactly as wide as the content (plus boxsep and boxrule). It comes with an optional argument and 4 mandatory arguments:
\begin{cverbbox}[options]{boxname}{text color}{box color}{frame color}
...
\end{cverbbox}
where the options are the same as already work in verbatimbox environments (in this MWE, I used the option to make the verbatim \tiny and added a bullet before each row). It uses \fboxsep and \fboxrule in the standard manner to set frame offset and rule thickness.
\documentclass{article}
\usepackage{xcolor}
\usepackage{verbatimbox}
\makeatletter
\newenvironment{cverbbox}[5][]{%
\setcounter{VerbboxLineNo}{0}%
\def\verbatim@processline{%
% THE FIRST #1 ACCOUNTS FOR NON-PRINTING COMMANDS; THE SECOND #1 IS FOR
% PRINTED OPTIONAL MATERIAL
{\addtocounter{VerbboxLineNo}{1}%
#1\setbox0=\hbox{#1\the\verbatim@line}%
\hsize=\wd0 \the\verbatim@line\par}}%
\@minipagetrue%
\@tempswatrue%
\global\edef\sv@name{\@macro@name{#2}}%
\global\edef\cverbboxColor{#4}%
\global\edef\cverbboxFColor{#5}%
\@ifundefined{\sv@name content}{%
\expandafter\newsavebox\expandafter{\csname\sv@name content\endcsname}%
}%
\expandafter\global\expandafter\edef\csname\sv@name\endcsname{\usebox{%
\csname\sv@name content\endcsname}}%
\setbox0=\vbox\bgroup\color{#3} \verbatim
}
{%
\endverbatim%
\unskip\setbox0=\lastbox %
\egroup%
\setbox1=\hbox{%
\colorbox{\cverbboxColor}{\box0}}%
\global\sbox{\csname\sv@name content\endcsname}%
{\fboxsep=\fboxrule\colorbox{\cverbboxFColor}{\box1}}%
}
\makeatother
\fboxrule=1pt\fboxsep=3pt\relax
\begin{document}
\begin{cverbbox}[\tiny\textcolor{black}{$\bullet$}]{\mycvbox}{red!80}{blue!10}{cyan}
\verbatim <Text>
Here
%$#@&^* \macros
xa
\end{cverbbox}
x\fbox{a}\mycvbox\fbox{b}x
\end{document}

fancyvrb, you need to use\VerbatimEnvironmentbefore the\begin{Verbatim}. That tellsfancyvrbto detect the name of the current environment, and look for the end of that environment, rather than looking for a literal\end{Verbatim}. – G. Poore Apr 15 '14 at 20:10tcolorbox– egreg Apr 15 '14 at 20:37