  We report the discovery of the millisecond pulsar (MSP) PSR J1950+2414
($P=4.3$ ms) in a binary system with an eccentric ($e=0.08$) 22-day orbit in
Pulsar ALFA survey observations with the Arecibo telescope. Its companion star
has a median mass of 0.3 $M_\odot$ and is most likely a white dwarf. Fully
recycled MSPs like this one are thought to be old neutron stars spun-up by mass
transfer from a companion star. This process should circularize the orbit, as
is observed for the vast majority of binary MSPs, which predominantly have
orbital eccentricities $e < 0.001$. However, four recently discovered binary
MSPs have orbits with $0.027 < e < 0.44$; PSR J1950+2414 is the fifth such
system to be discovered. The upper limits for its intrinsic spin period
derivative and inferred surface magnetic field strength are comparable to those
of the general MSP population. The large eccentricities are incompatible with
the predictions of the standard recycling scenario: something unusual happened
during their evolution. Proposed scenarios are a) initial evolution of the
pulsar in a triple system which became dynamically unstable, b) origin in an
exchange encounter in an environment with high stellar density, c) rotationally
delayed accretion-induced collapse of a super-Chandrasekhar white dwarf, and d)
dynamical interaction of the binary with a circumbinary disk. We compare the
properties of all five known eccentric MSPs with the predictions of these
formation channels. Future measurements of the masses and proper motion might
allow us to firmly exclude some of the proposed formation scenarios.
