  In this paper, we provide conditions, \emph{based solely on the observed
data}, for the global well-posedness, regularity and convergence of the
Azouni-Olson-Titi data assimilation algorithm (AOT algorithm) for a Leray-Hopf
weak solutions of the three dimensional Navier-Stokes equations (3D NSE). The
aforementioned conditions on the observations, which in this case comprise
either of \emph{modal} or \emph{volume element observations}, are automatically
satisfied for solutions that are globally regular and are uniformly bounded in
the $H^1$-norm. However, neither regularity nor uniqueness is necessary for the
efficacy of the AOT algorithm. To the best of our knowledge, this is the first
such rigorous analysis of the AOT data assimilation algorithm for the 3D NSE.
