Let $K_ℓ^-$ denote the graph obtained from $K_ℓ$ by deleting one edge. We show that for every $γ >0$ and every integer $ℓ≥4$ there exists an integer $n_0=n_0(γ ,ℓ)$ such that every graph $G$ whose order $n≥n_0$ is divisible by $ℓ$ and whose minimum degree is at least $(\frac{ℓ^2-3ℓ+1}{/ ℓ(ℓ-2)}+γ )n$ contains a $K_ℓ^-$-factor, i.e. a collection of disjoint copies of $K_ℓ^-$ which covers all vertices of $G$. This is best possible up to the error term $γn$ and yields an approximate solution to a conjecture of Kawarabayashi.