From reading the comments, this law's actually been in the books since 1997. Some others who said they actually read that law are saying that solemnization as referred to in the law doesn't prohibit churches from performing ceremonies, but filing actual paperwork to get a marriage license in the state.
Do churches actually do that? I always thought that people would have the ceremony in the church (which is more or less meaningless in the eyes of the state), and then they file for a marriage certificate themselves. But either way, the hypocrisy of this law is colossal.
Most priests are given the rights by the state to sign marriage licenses. In general most people when they get married do the whole ceremony in the church and then between the ceremony and the reception they sign the papers. The priest and the couple as well as their witnesses (normally first man and first bridesmaid) move off to the side of the church where they sign the document. This normally happens when the rest of the visitors are filing out of the church towards the reception. Of course, some couples make the signing part a central aspect of the ceremony.
What gays could do is exactly as you state though, they can have a ceremony that is essentially meaningless in the eyes of the state and then fill in the forms with a state registrar. They could even bring that person along for the ceremony. This law however prevents the priests from acting in their capacity as a recognized registrar for gay marriages.
Most countries that have legalized same sex marriage have included in the law an option for priests to opt-out of their duties as a registrar for same sex marriages, however state employees do not have such a luxury.
In short this law is pretty damn stupid.