  We develop the ultraspherical rectangular collocation (URC) method, a
collocation implementation of the sparse ultraspherical method of Olver \&
Townsend for two-point boundary-value problems. The URC method is provably
convergent, the implementation is simple and efficient, the convergence proof
motivates a preconditioner for iterative methods, and the modification of
collocation nodes is straightforward. The convergence theorem applies to all
boundary-value problems when the coefficient functions are sufficiently smooth
and when the roots of certain ultraspherical polynomials are used as
collocation nodes. We also adapt a theorem of Krasnolsel'skii et al.~to our
setting to prove convergence for the rectangular collocation method of Driscoll
\& Hale for a restricted class of boundary conditions.
