Summary: An orthogonal double cover (ODC) of

${K}_{n}$ is a collection of graphs such that each edge of

${K}_{n}$ occurs in exactly two of the graphs and two graphs have precisely one edge in common. ODCs of

${K}_{n}$ and their generalizations have been extensively studied by several authors (e.g. in

*J. H. Dinitz* and

*D. R. Stinson* (eds.) [Contemporary design theory (Wiley, New York) (1992;

Zbl 0746.00028), pp. 13–40 (Chapter 2)];

*H.-J. O. F. Gronau* et al. [Des. Codes Cryptography 27, 49–91 (2002;

Zbl 1001.05091)];

*H.-J. O. F. Gronau* et al. [Graphs Comb. 13, 251–262 (1997;

Zbl 0885.05093)];

*V. Leck* [Orthogonal double covers of

${K}_{m}$, Ph.D. Thesis, Universität Rostock (2000)]). In this paper, we investigate ODCs where the graph to be covered twice is

${K}_{n,n}$ and all graphs in the collection are isomorphic to a given small graph

$G$. We prove that there exists an ODC of

${K}_{n,n}$ by all proper subgraphs

$G$ of

${K}_{n,n}$ for

$1\u2a7dn\u2a7d9$, with two genuine exceptions.