The accretion-powered millisecond pulsar IGR J00291+5934 underwent two ~10 d long outbursts during 2008, separated by 30 d in quiescence. Such a short quiescent period between outbursts has never been seen before from a neutron star X-ray transient. X-ray pulsations at the 599 Hz spin frequency are detected throughout both outbursts. For the first time, we derive a pulse phase model that connects two outbursts, providing a long baseline for spin frequency measurement. Comparison with the frequency measured during the 2004 outburst of this source gives a spin-down during quiescence of -4(1)x10^-15 Hz/s, approximately an order of magnitude larger than the long-term spin-down observed in the 401 Hz accretion-powered pulsar SAX J1808.4-3658. If this spin-down is due to magnetic dipole radiation, it requires a 2x10^8 G field strength, and its high spin-down luminosity may be detectable with the Fermi Large Area Telescope. Alternatively, this large spin-down could be produced by gravitational wave emission from a fractional mass quadrupole moment of Q/I = 1x10^{-9}. The rapid succession of the outbursts also provides a unique test of models for accretion in low-mass X-ray binaries. Disk instability models generally predict that an outburst will leave the accretion disk too depleted to fuel a second outburst after such a brief quiescence. We suggest a modification in which the outburst is shut off by the onset of a propeller effect before the disk is depleted. This model can explain the short quiescence and the unusually slow rise of the light curve of the second 2008 outburst.