03/08/2007 at 02:00 PM EST
Johnny Depp's daughter, who has reportedly fallen ill in England, is "doing much better" the actor's rep said in a statement Thursday.
"We are happy to report that their daughter is doing much better," Depp's rep, Robin Baum, said in a statement. "The family greatly appreciates the continued support and respect of their privacy."
Depp and his longtime girlfriend , Vanessa Paradis, have two children, Lily-Rose, 7, and Jack, 5.
Several British papers have reported that Lily-Rose came down with an undisclosed illness over a week ago and that Depp, 43, has been by her bedside every day.
Depp has been filming a movie version of the stage musical Sweeney Todd for DreamWorks. A studio rep tells PEOPLE: "We have adjusted the schedule to accommodate [Depp's] needs at this moment. We understand that [Lily-Rose] is improving. ... Obviously everybody at the studio is certainly with them in their good spirits."