\documentclass[12pt,a4paper]{article}
\usepackage{fixltx2e}%% the official fixes for LaTeX2e
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}
\usepackage[scaled=0.92]{helvet}
\usepackage{mathptmx}
\usepackage{courier}
\newcounter{myEnumCounter}
\newcounter{mySaveCounter}
\renewenvironment{enumerate}{%
  \begin{list}{\arabic{myEnumCounter}.}{\usecounter{myEnumCounter}%
  \setcounter{myEnumCounter}{\value{mySaveCounter}}}%
}{\setcounter{mySaveCounter}{\value{myEnumCounter}}%
  \end{list}}
\newcommand\myEnumReset{\setcounter{mySaveCounter}{0}}
\makeatother
\begin{document}

\section{the first section}
\begin{enumerate}
\item eine zeile
\item eine Zeile
\end{enumerate}
another text with no sense another text with no sense another text
with no sense another text with no sense another text with no sense
another text with no sense another text with no sense 

\section{New section}
\begin{enumerate}
\item a line
\item another line
\end{enumerate}
now I'm writing \verb|\myEnumReset| \myEnumReset
and the counting starts new. 
another text with no sense another text with no sense another text
with no sense another text with no sense another text with no sense
another text with no sense another text with no sense 

\begin{enumerate}
\item another line
\item a line
\end{enumerate}
another text with no sense another text with no sense another text
with no sense another text with no sense another text with no sense 
\end{document}