In c++, why does the compiler choose the non-const function when the const would work also?

class Foo
{
public:
    std::string& Name()
    {
        m_maybe_modified = true;
        return m_name;
    }

    const std::string& Name() const
    {
        return m_name;
    }
protected:
    std::string m_name;
    bool m_maybe_modified;
};

Foo *a;
// Do stuff...
std::string name = a->Name(); // <-- chooses the non-const version

const Foo* b = a;
std::string name = b->Name();

std::string name = b->Name();

b->Name() = "me";