  [abridged] Although gravitational radiation causes inspiralling compact
binaries to circularize, a variety of astrophysical scenarios suggest that
binaries might have small but nonnegligible orbital eccentricities when they
enter the low-frequency bands of ground and space-based gravitational-wave
detectors. If not accounted for, even a small orbital eccentricity can cause a
potentially significant systematic error in the mass parameters of an
inspiralling binary. Gravitational-wave search templates typically rely on the
quasi-circular approximation, which provides relatively simple expressions for
the gravitational-wave phase to 3.5 post-Newtonian (PN) order. The
quasi-Keplerian formalism provides an elegant but complex description of the
post-Newtonian corrections to the orbits and waveforms of inspiralling binaries
with any eccentricity. Here we specialize the quasi-Keplerian formalism to
binaries with low eccentricity. In this limit the non-periodic contribution to
the gravitational-wave phasing can be expressed explicitly as simple functions
of frequency or time, with little additional complexity beyond the well-known
formulas for circular binaries. These eccentric phase corrections are computed
to 3PN order and to leading order in the eccentricity for the standard PN
approximants. For a variety of systems these eccentricity corrections cause
significant corrections to the number of gravitational wave cycles that sweep
through a detector's frequency band. This is evaluated using several measures,
including a modification of the useful cycles. We also evaluate the role of
periodic terms that enter the phasing and discuss how they can be incorporated
into some of the PN approximants. While the eccentric extension of the PN
approximants is our main objective, this work collects a variety of results
that may be of interest to others modeling eccentric relativistic binaries.
