Compare two files, then overwrite first file with only that in both files

I want to compare two files, and search for items that are in both. Then override the first file with that containing only elements which were in both files. I imagine something with diff, but not sure.

File 1

One
Two
Three
Four
Five

File 2

One
Three
Four
Six
Eight

Output (the new File 1)

One
Three
Four

finding the common items can be done using two ways

fgrep -f file1 file2

or

awk 'NR==FNR{a[$1]=$1;next} a[$1]' file1 file2

Join

should be helpful for you!