Downcasting shared_ptr to shared_ptr?

Update: the shared_ptr in this example is like the one in Boost, but it doesn't support shared_polymorphic_downcast (or dynamic_pointer_cast or static_pointer_cast for that matter)!

I'm trying to initialize a shared pointer to a derived class without losing the reference count:

struct Base { };
struct Derived : public Base { };
shared_ptr<Base> base(new Base());
shared_ptr<Derived> derived;

// error: invalid conversion from 'Base* const' to 'Derived*'
derived = base;  

So far, so good. I didn't expect C++ to implicitly convert Base* to Derived*. However, I do want the functionality expressed by the code (that is, maintaining the reference count while downcasting the base pointer). My first thought was to provide a cast operator in Base so that an implicit conversion to Derived could take place (for pedants: I would check that the down cast is valid, don't worry):

struct Base {
  operator Derived* ();
}
// ...
Base::operator Derived* () {
  return down_cast<Derived*>(this);
}

Well, it didn't help. It seems the compiler completely ignored my typecast operator. Any ideas how I could make the shared_ptr assignment work? For extra points: what kind of type Base* const is? const Base* I understand, but Base* const? What does const refer to in this case?