  The measurement and stabilization of the carrier-envelope offset frequency
$f_{\textrm{CEO}}$ via self-referencing is paramount for optical frequency comb
generation which has revolutionized precision frequency metrology,
spectroscopy, and optical clocks. Over the past decade, the development of
chip-scale platforms has enabled compact integrated waveguides for
supercontinuum generation. However, there is a critical need for an on-chip
self-referencing system that is adaptive to different pump wavelengths,
requires low pulse energy, and does not require complicated processing. Here,
we demonstrate efficient carrier-envelope offset frequency $f_{\textrm{CEO}}$
stabilization of a modelocked laser with only 107 pJ of pulse energy via
self-referencing in an integrated lithium niobate waveguide. We realize an
$f$-$2f$ interferometer through second-harmonic generation and subsequent
supercontinuum generation in a single dispersion-engineered waveguide with a
stabilization performance equivalent to a conventional off-chip module. The
$f_{\textrm{CEO}}$ beatnote is measured over a pump wavelength range of 70 nm.
We theoretically investigate our system using a single nonlinear envelope
equation with contributions from both second- and third-order nonlinearities.
Our modeling reveals rich ultrabroadband nonlinear dynamics and confirms that
the initial second harmonic generation followed by supercontinuum generation
with the remaining pump is responsible for the generation of a strong
$f_{\textrm{CEO}}$ signal as compared to a traditional $f$-$2f$ interferometer.
Our technology provides a highly-simplified system that is robust, low cost,
and adaptable for precision metrology for use outside a research laboratory.
