A graph $G$ is said to be $\preceq$-ubiquitous, where $\preceq$ is the minor relation between graphs, if whenever $Γ$ is a graph with $nG \preceq Γ$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \preceq Γ$, where $αG$ is the disjoint union of $α$ many copies of $G$. A well-known conjecture of Andreae is that every locally finite connected graph is $\preceq$-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph~$G$ which implies that $G$ is $\preceq$-ubiquitous. In particular this implies that the full grid is $\preceq$-ubiquitous.