  The exchange of an axion-like particle between atomic electrons and the
nucleus may induce electric dipole moments (EDMs) of atoms and molecules. This
interaction is described by a parity- and time-reversal-invariance-violating
potential which depends on the product of a scalar $g^s$ and a pseudoscalar
$g^p$ coupling constant. We consider the interaction with the specific
combination of these constants, $g_e^s g_N^p$, which gives significant
contributions to the EDMs of diamagnetic atoms. In this paper, we calculate
these contributions to the EDMs of $^{199}$Hg, $^{129}$Xe, $^{211}$Rn and
$^{225}$Ra for a wide range of axion masses. Comparing these results with
recent experimental EDM measurements, we place new constraints on $g_e^s
g_N^p$. The most stringent atomic EDM limits come from $^{199}$Hg and improve
on existing laboratory limits from other experiments for axion masses exceeding
$10^{-2}$ eV.
