  We investigate magic-wavelength trapping of ultracold bialkali molecules in
the vicinity of weak optical transitions from the vibrational ground state of
the X$^1\Sigma^+$ potential to low-lying rovibrational states of the b$^3\Pi_0$
potential, focussing our discussion on the $^{87}$Rb$^{133}$Cs molecule in a
magnetic field of $B=181\,$G. We show that a frequency window exists between
two nearest neighbor vibrational poles in the dynamic polarizability where the
trapping potential is "near magic" for multiple rotational states
simultaneously. We show that the addition of a modest DC electric field of
$E=0.13\,\text{kV}/\text{cm}$ leads to an exact magic-wavelength trap for the
lowest three rotational states at a angular-frequency detuning of
$\Delta_{v'=0} = 2\pi\times 218.22$\,GHz from the X$^1\Sigma^+ (v=0,
J=0)\rightarrow$ b$^3\Pi_0 (v'=0, J=1)$ transition. We derive a set of
analytical criteria that must be fulfilled to ensure the existence of such
magic frequency windows and present an analytic expression for the position of
the frequency window in terms of a set of experimentally measurable parameters.
These results should inform future experiments requiring long coherence times
on multiple rotational transitions in ultracold polar molecules.
