Warning: This does not consider newlines. For a more in-depth answer, see this SO-question instead. (Thanks, Ed Morton & Niklas Peter)
Note that escaping everything is a bad idea. Sed needs many characters to be escaped to get their special meaning. For example, if you escape a digit in the replacement string, it will turn in to a backreference.
As Ben Blank said, there are only three characters that need to be escaped in the replacement string (escapes themselves, forward slash for end of statement and & for replace all):
ESCAPED_REPLACE=$(printf '%s\n' "$REPLACE" | sed -e 's/[\/&]/\\&/g')
# Now you can use ESCAPED_REPLACE in the original sed statement
sed "s/KEYWORD/$ESCAPED_REPLACE/g"
If you ever need to escape the KEYWORD string, the following is the one you need:
sed -e 's/[]\/$*.^[]/\\&/g'And can be used by:
KEYWORD="The Keyword You Need";
ESCAPED_KEYWORD=$(printf '%s\n' "$KEYWORD" | sed -e 's/[]\/$*.^[]/\\&/g');
# Now you can use it inside the original sed statement to replace text
sed "s/$ESCAPED_KEYWORD/$ESCAPED_REPLACE/g"
Remember, if you use a character other than / as delimiter, you need replace the slash in the expressions above wih the character you are using. See PeterJCLaw’s comment for explanation.
Edited: Due to some corner cases previously not accounted for, the commands above have changed several times. Check the edit history for details.