  We show that for any weakly convergent sequence of ergodic
$SL_2(\mathbb{R})$-invariant probability measures on a stratum of unit-area
translation surfaces, the corresponding Siegel-Veech constants converge to the
Siegel-Veech constant of the limit measure. Together with a measure
equidistribution result due to Eskin-Mirzakhani-Mohammadi, this yields the
(previously conjectured) convergence of sequences of Siegel-Veech constants
associated to Teichm\"uller curves in genus two.
  The proof uses a recurrence result closely related to techniques developed by
Eskin-Masur. We also use this recurrence result to get an asymptotic quadratic
upper bound, with a uniform constant depending only on the stratum, for the
number of saddle connections of length at most $R$ on a unit-area translation
surface.
