  We report on the spectroscopy of radio-frequency transitions between
nearly-degenerate, opposite-parity excited states in atomic dysprosium (Dy).
Theoretical calculations predict that these states are very sensitive to
variation of the fine-structure constant, $\alpha$, owing to large relativistic
corrections of opposite sign for the opposite-parity levels. The near
degeneracy reduces the relative precision necessary to place constraints on
variation of $\alpha$ competitive with results obtained from the best atomic
clocks in the world. Additionally, the existence of several abundant isotopes
of Dy allows isotopic comparisons that suppress common-mode systematic errors.
The frequencies of the 754-MHz transition in $^{164}$Dy and 235-MHz transition
in $^{162}$Dy were measured over the span of two years. Linear variation of
$\alpha$ is found to be $\dot{\alpha}/\alpha = (-5.8\pm6.9)\times10^{-17}$
yr$^{-1}$, consistent with zero. The same data are used to constrain the
dimensionless parameter $k_\alpha$, characterizing a possible coupling of
$\alpha$ to a changing gravitational potential. We find that $k_\alpha =
(-5.5\pm5.2)\times10^{-7}$, essentially consistent with zero and the best
constraint to date.
