  We construct a minimal mass blow up solution of the modified Benjamin-Ono
equation (mBO) \[ u_{t}+(u^3-D^1 u)_{x}=0, \] which is a standard mass critical
dispersive model. Let $Q\in H^{\frac 12}$, $Q>0$, be the unique ground state
solution of $D^1 Q +Q=Q^3$, constructed using variational arguments by
Weinstein (Comm. PDE, 12 (1987), J. Diff. Eq., 69 (1987)) and Albert, Bona and
Saut (Proc. Royal London Soc., 453 (1997)), and whose uniqueness was recently
proved by Frank and Lenzmann (Acta Math., 210 (2013)).
  We show the existence of a solution $S$ of (mBO) satisfying $\|S
\|_{L^2}=\|Q\|_{L^2}$ and \[ S(t)-\frac1{\lambda^{\frac12}(t)}
Q\left(\frac{\cdot - x(t)}{\lambda(t)}\right)\to 0\quad \mbox{ in }\ H^{\frac
12}(\mathbb R) \mbox{ as }\ t\downarrow 0, \] where \[ \lambda(t)\sim t,\quad
x(t) \sim -|\ln t| \quad \hbox{and}\quad \|S(t)\|_{\dot H^{\frac 12}} \sim
t^{-\frac 12}\|Q\|_{\dot H^{\frac 12}} \quad \hbox{as}\ t\downarrow 0. \] This
existence result is analogous to the one obtained by Martel, Merle and
Rapha\"el (J. Eur. Math. Soc., 17 (2015)) for the mass critical generalized
Korteweg-de Vries equation. However, in contrast with the (gKdV) equation, for
which the blow up problem is now well-understood in a neighborhood of the
ground state, $S$ is the first example of blow up solution for (mBO).
  The proof involves the construction of a blow up profile, energy estimates as
well as refined localization arguments, developed in the context of
Benjamin-Ono type equations by Kenig, Martel and Robbiano (Ann. Inst. H.
Poincar\'e, Anal. Non Lin., 28 (2011)). Due to the lack of information on the
(mBO) flow around the ground state, the energy estimates have to be
considerably sharpened in the present paper.
