  The simplest molecules in nature, molecular hydrogen ions in the form of H2+
and HD+, provide an important benchmark system for tests of quantum
electrodynamics in complex forms of matter. Here, we report on such a test
based on a frequency measurement of a vibrational overtone transition in HD+ by
laser spectroscopy. We find that the theoretical and experimental frequencies
are equal to within 0.6(1.1) parts per billion, which represents the most
stringent test of molecular theory so far. Our measurement not only confirms
the validity of high-order quantum electrodynamics in molecules, but also
enables the long predicted determination of the proton-to-electron mass ratio
from a molecular system, as well as improved constraints on hypothetical fifth
forces and compactified higher dimensions at the molecular scale. With the
perspective of comparisons between theory and experiment at the 0.01
part-per-billion level, our work demonstrates the potential of molecular
hydrogen ions as a probe of fundamental physical constants and laws.
