AbstractCellular covers of groups, and in particular, those of divisible abelian groups, were studied in [FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of groups, J. Pure Appl. Algebra 208, (2007), 61–76], [CHA-CHÓLSKI, W.—FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of divisible abelian groups. In: Contemp. Math. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 77–97], and continued in [FUCHS, L.—GÖBEL, R.: Cellular covers of abelian groups, Results Math. 53, (2009), 59–76] for abelian groups in general. In this note we are investigating cellular covers in the category of totally ordered abelian groups (called o-cellular covers; for definition see Section 2). Some results are similar to those on torsion-free abelian groups (unordered), while others are completely different. For instance, though kernels of o-cellular covers can not be non-zero divisible groups (Lemma 3.1), they may contain non-zero divisible subgroups (Example 3.2); however, the divisible part can not be much larger than the reduced part (Theorem 3.4). There are o-groups, even among the additive subgroups of the rationals, whose o-cellular covers form a proper class (Theorem 4.3).