Request root privilege from within a script

This'll work:

echo "$(whoami)"

[ "$UID" -eq 0 ] || exec sudo "$0" "$@"

example:

./test.sh 
blade
[sudo] password for blade: 
root

If you'd like a pretty dialog, try something like this. I ripped this straight out of something else I wrote, so it's got extra stuff you might not need or want, but it shows the general idea:

brand="My Software"

# Check that the script is running as root. If not, then prompt for the sudo
# password and re-execute this script with sudo.
if [ "$(id -nu)" != "root" ]; then
    sudo -k
    pass=$(whiptail --backtitle "$brand Installer" --title "Authentication required" --passwordbox "Installing $brand requires administrative privilege. Please authenticate to begin the installation.\n\n[sudo] Password for user $USER:" 12 50 3>&2 2>&1 1>&3-)
    exec sudo -S -p '' "$0" "$@" <<< "$pass"
    exit 1
fi

This uses whiptail, which you can install if you don't already have it:

sudo apt-get install whiptail

blade19899's answer is indeed the way to go, however one could also call sudo bash in the shebang:

#!/usr/bin/sudo bash
# ...

The obvious caveat is this will work only as long as the script is called with ./script and will fail as soon as the script is called with bash script.