(CNN) The ACLU of Ohio, Planned Parenthood and other groups are suing the state of Ohio over its new law banning abortions after a fetal heartbeat can be detected.

The plaintiffs want a federal court to block the measure before it takes effect on July 10.

In a teleconference, Elizabeth Watson, a staff attorney at the ACLU's Reproductive Freedom Project, said she believed the law was "blatantly unconstitutional under more than 45 years of Supreme Court precedent."

She said the ban at six weeks constitutes a prohibition "on almost all abortions" because it's a stage so early that "most people do not even know that they are pregnant."

Watson said that abortion is under unprecedented attack despite "strong public support for supportive, affordable, and barrier-free access to abortion care."

In a statement, Dan Tierney, a spokesman for Ohio Gov. Mike DeWine, told CNN, "This is an expected development. Governor DeWine has long said that this issue will ultimately be decided by the United States Supreme Court."

DeWine signed the bill into law on April 11.

At the time, Ohio Attorney General David Yost touted the law, saying, "In the last 46 years, the practice of medicine has changed. Science has changed. Even the point of viability has changed. Only the law has lagged behind. This law provides a stable, objective standard to guide the courts."

Groups hope to prevent the law from taking effect

Plaintiffs are Preterm-Cleveland, Planned Parenthood Greater Ohio, Planned Parenthood Southwest Ohio Region, Women's Med Corp Professional Group, Capital Care Network of Toledo and Sharon Liner M.D.

The suit names various state and local officials as defendants, including Yost, the director of the state's Department of Health, leaders in the State Medical Board of Ohio and several county prosecutors.

The suit seeks a temporary restraining order or preliminary injunction before the Ohio law takes effect.

The attorneys argue that the state's new abortion law "is a direct violation of Plaintiffs' patients' fundamental constitutional right to decide whether to have an abortion prior to viability, and causes those patients irreparable harm."

A fetal heartbeat can be detected at about the six-week mark in a woman's pregnancy.

Heartbeat bills are stacking up, and so are lawsuits against them

Mississippi, Kentucky, Arkansas and Georgia also have passed "heartbeat bills" this year. Alabama's legislature passed the most restrictive anti-abortion law in the country. That bill now heads the governor's desk.

The ACLU sued against the Kentucky law, and it was blocked by the court. A lawsuit is pending in Mississippi. In Georgia, Watson, the ACLU lawyer, said that "we, along with others, have vowed to challenge it before it takes effect."

Watson promised similar actions if Alabama Gov. Kay Ivey signs the bill on her desk this week.