Bugzilla import fails: "IllegalArgumentException: Passed Collection with no elements"

When the Bugzilla importer gets as far as importing issues, it dies with an exception:

Error importing data from bugzilla.java.lang.IllegalArgumentException: Passed Collection with no elements
at com.atlassian.jira.util.BugzillaImportBean.getOnly(BugzillaImportBean.java:781)
at com.atlassian.jira.util.BugzillaImportBean.setCurrentWorkflowStep(BugzillaImportBean.java:296)
at com.atlassian.jira.util.BugzillaImportBean.createIssue(BugzillaImportBean.java:250)
at com.atlassian.jira.util.BugzillaImportBean.createIssues(BugzillaImportBean.java:206)
at com.atlassian.jira.util.BugzillaImportBean.create(BugzillaImportBean.java:146)
at com.atlassian.jira.web.action.util.BugzillaImport.doRun(BugzillaImport.java:118)