  We investigate the possibility of viable leptogenesis in an appealing
$\Delta(27)$ model with a universal texture zero in the (1,1) entry. The model
accommodates the mass spectrum, mixing and CP phases for both quarks and
leptons and allows for grand unification. Flavoured Boltzmann equations for the
lepton asymmetries are solved numerically, taking into account both $N_1$ and
$N_2$ right-handed neutrino decays. The $N_1$-dominated scenario is successful
and the most natural option for the model, with $M_1 \in [10^9, 10^{12}]$ GeV,
and $M_1/M_2 \in [0.002, 0.1]$, which constrains the parameter space of the
underlying model and yields lower bounds on the respective Yukawa couplings.
Viable leptogenesis is also possible in the $N_2$-dominated scenario, with the
asymmetry in the electron flavour protected from $N_1$ washout by the texture
zero. However, this occurs in a region of parameter space which has a stronger
mass hierarchy $M_1/M_2 < 0.002 $, and $M_2$ relatively close to $M_3$, which
is not a natural expectation of the $\Delta(27)$ model.
