cat and grep without destroying file

I've been trying to make a small shell script that would remove a line from a text file. But every time that I pipe the content of grep to the original file, the file ends up blank. So, I've made the script use a copy and them replace the original. It looks like this:
#for i in $(/bin/ls -1); do cat $i | egrep -v ^Date > $i.new; mv $i.new $i; done

Isn't there any other way, without using the mv?

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