To obtain a p-value, you need to compare with some distribution and a chi-square with one df is the default output. Often however a mixture of 0 and 1 df's are more appropriate, hence a more correct p-value is half the one, the software reports.
For linear mixed models with uncorrelated random effects, our package RLRsim offers a rapid algorithm to determine the exact finite sample distribution of the restricted likelihood ratio for testing whether the variance of a random effect is zero. Using the ChiSquare(1) or a 50:50 mixture of ChiSquare(1) and 0 will almost always lead to very conservative tests. For a detailed comparison of various approaches to test for zero variance in linear mixed models have a look at: F.Scheipl, S.Greven, H.K?chenhoff (2008): Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models. Computational Statistics & Data Analysis, 52(7):3283-3299 (http://dx.doi.org/10.1016/j.csda.2007.10.022).