  In this paper, we study weakly unital dg categories as they were defined by
Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated
Quillen model structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$
of small weakly unital dg categories over a field $\Bbbk$. Our model structure
can be thought of as an extension of the model structure on the category
$\mathrm{Cat}_{\mathrm{dg}}(\Bbbk)$ of (strictly unital) small dg categories
over $\Bbbk$, due to Tabuada [Tab]. More precisely, we show that the imbedding
of $\mathrm{Cat}_{\mathrm{dg}}(\Bbbk)$ to $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$
is a right adjoint of a Quillen pair of functors. We prove that this Quillen
pair is, in turn, a Quillen equivalence. In course of the proof, we study a
non-symmetric dg operad $\mathcal{O}$, governing the weakly unital dg
categories, which is encoded in the Kontsevich-Soibelman definition. We prove
that this dg operad is quasi-isomorphic to the operad $\mathrm{Assoc}_+$ of
unital associative algebras.
