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procedure TForm1.Button1Click(Sender: TObject);
var gb:textfile;
begin
assignfile(gb,'C:\Dokumente und Einstellungen\Stefanie\Desktop\Delphi Projektordner\Gästebuch\gb.txt');
{$I-}reset(gb); {$I+};
if ioresult<>0 then rewrite(gb);
write(gb,Memo1.text);                          //hier bleibt es hängen
closefile(gb);					

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procedure TForm1.Button1Click(Sender: TObject);
const
  FileName = 'C:\Gaestebuch.txt';
var
  Stream: THandleStream;
  Handle: Integer;
begin

  if FileExists(FileName) then
    Handle := FileOpen(FileName, fmOpenReadWrite)
  else
    Handle := FileCreate(FileName);

  if Handle <> -1 then
  begin
    Stream := THandleStream.Create(Handle);
    try
      Stream.Seek(0, soFromEnd);
      Memo1.Lines.SaveToStream(Stream);
    finally
      Stream.Free;
      CloseHandle(Handle);
    end;
  end;

end;					

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Memo1.lines.SavetoFile('C:\Dokumente und Einstellungen\Stefanie\Desktop\Delphi Projektordner\Gästebuch\gb.txt');					

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AssignFile(F,...)
try
  try 
    Append(F);
  except
    Rewrite(F)
  end;
  ...
except
  CloseFile(F);
end;