  For given positive integers $k$ and $n$, a family $\mathcal{F}$ of subsets of
$\{1,\dots,n\}$ is $k$-antichain saturated if it does not contain an antichain
of size $k$, but adding any set to $\mathcal{F}$ creates an antichain of size
$k$. We use sat$^*(n, k)$ to denote the smallest size of such a family. For all
$k$ and sufficiently large $n$, we determine the exact value of sat$^*(n, k)$.
Our result implies that sat$^*(n, k)=n(k-1)-\Theta(k\log k)$, which confirms
several conjectures on antichain saturation. Previously, exact values for
sat$^*(n,k)$ were only known for $k$ up to $6$.
  We also prove a generalisation of a result of Lehman-Ron which may be of
independent interest. We show that given $m$ disjoint chains in the Boolean
lattice, we can create $m$ disjoint skipless chains that cover the same
elements (where we call a chain skipless if any two consecutive elements differ
in size by exactly one).
