The copy constructor is provided whether you define any other constructors or not. As long as you don't declare a copy constructor, you get one.
The no-arg constructor is only provided if you declare no constructors. So you don't have a problem unless you want a no-arg constructor, but consider it a waste of time writing one.
IIRC, C++0x has a way of delegating construction to another constructor. I can't remember the details, but it would allow you to define a no-arg constructor by specifying another constructor, plus the argument(s) to pass to it. Might save typing some data member initializers in some cases. But the default no-arg constructor wouldn't have provided those initializers either.