  We compute the Darwin operator contribution ($1/m_b^3$ correction) to the
width of the inclusive non-leptonic decay of a $B$ meson ($B^+$, $B_d$ or
$B_s$), stemming from the quark flavour-changing transition $b \to q_1 \bar q_2
q_3$, where $q_1,q_2 = u, c$ and $q_3 = d, s$. The key ideas of the computation
are the local expansion of the quark propagator in the external gluon field
including terms with covariant derivative of the gluon field strength tensor
and the standard technique of the Heavy Quark Expansion (HQE). We confirm the
previously known expressions of the $1/m_b^3$ contributions to the
semi-leptonic decay $b \to q_1 \ell \bar \nu_\ell$, with $\ell = e, \mu, \tau$
and of the $1/m_b^2$ contributions to the non-leptonic modes. We find that this
new term can give a sizeable correction of about $- 4 \, \%$ to the
non-leptonic decay width of a $B$ meson. For $B_d$ and $B_s$ mesons this turns
out to be the dominant correction to the free b-quark decay, while for the
$B^+$ meson the Darwin term gives the second most important correction -
roughly 1/2 to 1/3 of the phase space enhanced Pauli interference contribution.
Due to the tiny experimental uncertainties in lifetime measurements the
incorporation of the Darwin term contribution is crucial for precision tests of
the Standard Model.
