This might work for you (GNU sed):
sed -ne '/Start/,/End/w outputfile' -e '/End/q' file
This will write to outputfile
the lines between the first Start
and End
and then quit and obviate the need to use grep too.
If you must use grep then perhaps:
sed -n "$(grep -n "Start" file | cut -d: -f 1 | head -n 1),$(grep -n "End" file | cut -d: -f 1 | head -n 1)"'p' file