Logistic regression problems with zero variance data


I am working with binary data. My research question was whether different types of learning experiences would impact whether or not a child would produce a particular response (0 or 1) within each of four different tasks. My predictors are:
- condition (there were four learning conditions total, this was b/w subjects where children were placed in one of these conditions)
- age (continuous, but only looking at a particular age range)
- gender (categorical)
- spatial test (the score they received on a test of spatial skill, there were 13 items and I just calculated the proportion correct)

Logistic regression analyses have worked fine so far for most of the data. However, in one of my tasks, one of my conditions has a mean of 0 (everyone responded with 0). Because of this lack of variance, the logistic regression totally freaks out and can't make comparisons against this mean. What do I do for this particular analysis?

Elsewhere, I've read about an "empirical logit transformation" appropriate for data points that lack variance. This was not helpful, as it was meant for aggregated data (i.e., multiple trials, rather than my situation, where in a single task, they either produced a 0 or a 1).

Can anyone provide a simple solution or insight into solving this problem? I appreciate any help. If it helps, I'm using SPSS to carry out these analyses.