  We examine in a model-independent manner the measurements that can be
performed at B-factories with sensitivity to dark matter. If a singlet scalar,
pseudo-scalar, or vector is present and mediates the Standard Model - dark
matter interaction, it can mediate invisible decays of quarkonium states such
as the $\Upsilon$, $J/\Psi$, and $\eta$. Such scenarios have arisen in the
context of supersymmetry, extended Higgs sectors, solutions the supersymmetric
$\mu$ problem, and extra U(1) gauge groups from grand unified theories and
string theory. Existing B-factories running at the $\Upsilon(4S)$ can produce
lower $\Upsilon$ resonances by emitting an Initial State Radiation (ISR)
photon. Using a combination of ISR and radiative decays, the initial state of
an invisibly decaying quarkonium resonance can be tagged, giving sensitivity to
the spin and CP-nature of the particle that mediates standard model-dark matter
interactions. These measurements can discover or place strong constraints on
dark matter scenarios where the dark matter is approximately lighter than the
$b$-quark. For the decay chains $\Upsilon(nS) \to \pi^+ \pi^- \Upsilon(1S)$
(n=2,3) we analyze the dominant backgrounds and determine that with $400
fb^{-1}$ collected at the $\Upsilon(4S)$, the B-factories can limit
$BR(\Upsilon(1S) \to invisible) \lsim 0.1%$.
