SortedSet<T> need to use different sort and equality criteria

I have a class that was previously being used in a HashSet. This has now been changed so that the class is now used in a SortedSet, but the equality test no longer works as it did. I believe this it uses the CompareTo function for both sorting and comparing, and this is by design.

Anyone have any ideas, other than performing my own duplicate checks?

public sealed class DatedID : IEquatable<DatedID>, IComparable
{
    readonly DateTime _added;
    readonly int _ID;

    public DateTime Added
    {
        get { return _added; }
    }

    public int ID
    {
        get { return _ID; }
    }

    public DatedID(int id)
        : this(id, DateTime.Now) {}

    public DatedID(int id, DateTime added)
    {
        id.ThrowDefault("id");
        added.ThrowDefault("added");

        _ID = id;
        _added = added;
    }

    // Compare

    int IComparable.CompareTo(object obj)
    {
        var other = (DatedID)obj;

        // Newest => oldest
        return this.Added > other.Added ? -1 : this.Added < other.Added ? 1 : 0;
    }

    // Equals

    public bool Equals(DatedID other)
    {
        if (other == null) return false;

        return this.ID == other.ID;
    }

    public override bool Equals(object obj)
    {
        if (obj == null) return false;

        var di = obj as DatedID;

        return di == null ? false : Equals(di);
    }

    public override int GetHashCode()
    {
        return ID.GetHashCode();
    }
}