The problem is to obtain an exact p-value for the one-sample chi-square for small sample situations where the Normal approximation does not hold. This can be achieved by using a family of methods known as exact tests.
In particular for a one sample chi-square test, we can generate random samples (groups) of particular sizes conditional on the sum of these samples having the same total sample size as that of observed data (ie generating observations from a multinomial distribution). If this is done a large number of times, say 5000, the p-value is equal to the proportion of times that the observed chi-square value is greater than or equal to the chi-square value observed from the data. A consequence of this method is that a different p-value will be generated each time this procedure is run although the differences should be negligible provided each run comprises of a large enough number of simulated random samples e.g. 5000.
An alternative strategy is to evaluate all possible combinations of groups whose sizes sum to the total sample size of the observed data and again express the p-value as the proportion of times the computed chi-square values are greater than or equal to the one observed from the data.
Reference
Edgington E., Onghena P. (2007) Randomization tests. 4th edition. Blackwell Publishing.
(This text gives examples of computing p-values using exact tests for a variety of commonly occurring statistical tests in cases where there are small sample sizes)