Throwing an exception when building an object during a throw?

I have an exception class:

class MyException : public std::exception
{
public:
    MyException( char* message )
        : message_( message )
    {
        if( !message_ ) throw std::invalid_argument("message param must not be null");
    }
};

And on my cast site:

try {
    throw MyException( NULL );
}
catch( std::exception const& e ) {
    std::cout << e.what();
}

(the code has not been compiled, so please excuse any errors)

I am wondering what will happen when I exit the constructor when building due to another throw. I assume this is legal and catch will catch std::invalid_argumentand the previous exception ( MyException) will be ignored or discarded.

My goal with this design is to enforce invariants in my exception class. message_it should never be NULL, and I do not want the conditions to check if it is null in my overload what(), so I check them in the constructor and throw if they are invalid.

Is this correct or different from behavior?