The concept of the "Kondo box" describes a single spin, antiferromagnetically coupled to a quantum dot with a finite level spacing. Here, a Kondo box is formed in a carbon nanotube interacting with a localized electron. We investigate the spins of its first few eigenstates and compare them to a recent theory. In an "open" Kondo-box, strongly coupled to the leads, we observe a nonmonotonic temperature dependence of the nanotube conductance, which results from a competition between the Kondo-box singlet and the "conventional" Kondo state that couples the nanotube to the leads. © 2010 The American Physical Society.