Copy files listed in text file to new directory

I am trying to write a script that will copy all file listed in a text file (100s of file names) to a new directory
Assume script will run with main as current working directory and I know how many files/lines will be in List.txt

Im trying to work up a test script using this model
Contents of main

script.sh
sort < is a folder
List.txt
stuff.pdf
misc.rtf
junk.txt
thing.txt

Contents of List.txt

stuff.pdf
misc.rtf
thing.txt

Script so far (contests of script.sh)

#1/bin/bash

count =0
while [ $count -le 3 ]
do
    filename= sed -n '1p' List
    cp $filename sort
    count=$(($count + 1 ))
done

2 things 2 Do
(1)
count though lines with counter
something like

filename= sed -n '$countp' List

(2)
use the file name stored in $filename in copy command
right now its viewing sort as the source not the destination $filename is not being read in

Its a small script but any help would be appreciated

Even though you don't know how many lines are there in the List (file with filenames), you could do copying to the directory.

cat List.txt | while read line
do
cp $line <<desired_directory>>/
done

You can also try this from command line

Text_File=$( awk '{print $1}' texfilename)
cp $Text_File targetdirectory