The classical problem of magnetic stripe domain behavior in films and plates with uniaxial magnetic anisotropy is addressed. Exact analytical results are derived for the stripe domain widths as a function of applied perpendicular field H, in the regime where the domain period becomes large. The stripe period diverges as (H-c - H)(-1/2), where H-c is the critical (infinite period) field, an exact result confirming a previous conjecture. The magnetization approaches saturation as (H-c - H)(1/2), a behavior that compares excellently with experimental data obtained for a 4-mu m thick ferrite garnet film. The exact analytical solution provides a new basis for precise characterization of uniaxial magnetic films and plates, illustrated by a simple way to measure the domain wall energy. The mathematical approach is applicable for similar analysis of a wide class of systems with competing interactions where a stripe domain phase is formed.