In this paper, we study the singular limit $q\to \infty$ for non-negative weak solutions of a doubly nonlinear parabolic Cauchy problem with absorption. Under the assumptions $m>1$, $m(p-1)>1$ and $q>p(m+2)$, we first prove that the family $\left\{u^{\left(q\right)}\right\}$ is precompact, uniformly on compact subsets of ${\mathbb{R}}^N\times (0,\infty )$, and that the whole family converges to the unique bounded weak solution of the homogeneous problem with initial datum $v_0\left(x\right)=min\{u_0\left(x\right),1\}$. We then show that the limit operator ${\mathcal{T}}_t$ is given by the composition of the homogeneous semigroup with the truncation $u_0\mapsto min\{u_0,1\}$; it forms an order-preserving semigroup, satisfies the $L^1$-contraction property, and exhibits an initial layer precisely characterized by the excess mass ${(u_0-1)}_{+}$. Finally, we compare the singular limit with the homogeneous solution and give a sharp description of the corresponding convergence defect.