What causes JtaStatusHelper to throw a NullPointerException when using Spring, Hibernate 4, and JTA in JBoss EAP 6?

Issue

When using Spring 3, Hibernate 4, and JTA in JBoss EAP 6, an exception like the following is thrown:

caused by: null: java.lang.NullPointerException
    at org.hibernate.engine.transaction.internal.jta.JtaStatusHelper.getStatus(JtaStatusHelper.java:73) [hibernate-core-4.1.3.Final-redhat-1.jar:4.1.3.Final-redhat-1]
    at org.hibernate.engine.transaction.internal.jta.JtaStatusHelper.isActive(JtaStatusHelper.java:115) [hibernate-core-4.1.3.Final-redhat-1.jar:4.1.3.Final-redhat-1]
    at org.hibernate.engine.transaction.internal.jta.CMTTransaction.join(CMTTransaction.java:149) [hibernate-core-4.1.3.Final-redhat-1.jar:4.1.3.Final-redhat-1]
    at org.hibernate.ejb.AbstractEntityManagerImpl.joinTransaction(AbstractEntityManagerImpl.java:1215) [hibernate-entitymanager-4.1.3.Final-redhat-1.jar:4.1.3.Final-redhat-1]
    at org.hibernate.ejb.AbstractEntityManagerImpl.postInit(AbstractEntityManagerImpl.java:177) [hibernate-entitymanager-4.1.3.Final-redhat-1.jar:4.1.3.Final-redhat-1]
    at org.hibernate.ejb.EntityManagerImpl.<init>(EntityManagerImpl.java:89) [hibernate-entitymanager-4.1.3.Final-redhat-1.jar:4.1.3.Final-redhat-1]
    at org.hibernate.ejb.EntityManagerFactoryImpl.createEntityManager(EntityManagerFactoryImpl.java:179) [hibernate-entitymanager-4.1.3.Final-redhat-1.jar:4.1.3.Final-redhat-1]
    at org.hibernate.ejb.EntityManagerFactoryImpl.createEntityManager(EntityManagerFactoryImpl.java:174) [hibernate-entitymanager-4.1.3.Final-redhat-1.jar:4.1.3.Final-redhat-1]
    at org.springframework.orm.jpa.EntityManagerFactoryUtils.doGetTransactionalEntityManager(EntityManagerFactoryUtils.java:195) [spring-orm-3.0.7.RELEASE.jar:3.0.7.RELEASE]
    at org.springframework.orm.jpa.EntityManagerFactoryUtils.getTransactionalEntityManager(EntityManagerFactoryUtils.java:141) [spring-orm-3.0.7.RELEASE.jar:3.0.7.RELEASE]
    at org.springframework.orm.jpa.EntityManagerFactoryAccessor.getTransactionalEntityManager(EntityManagerFactoryAccessor.java:129) [spring-orm-3.0.7.RELEASE.jar:3.0.7.RELEASE]
    at org.springframework.orm.jpa.JpaTemplate.execute(JpaTemplate.java:173) [spring-orm-3.0.7.RELEASE.jar:3.0.7.RELEASE]
    at org.springframework.orm.jpa.JpaTemplate.execute(JpaTemplate.java:146) [spring-orm-3.0.7.RELEASE.jar:3.0.7.RELEASE]