  In this paper, we introduce a new notion of biprojectivity, called
Connes-biprojective, for dual Banach algebras. We study the relation between
this new notion to Connes-amenability and we show that, for a given dual Banach
algebra $ \mathcal{A} $, it is Connes-amenable if and only if $ \mathcal{A} $
is Connes-biprojective and has a bounded approximate identity. Also, for an
Arens regular Banach algebra $ \mathcal{A} $, we show that if $ \mathcal{A} $
is biprojective, then the dual Banach algebra $ \mathcal{A} ^{**} $ is
Connes-biprojective.
