Demand for tickets, which cost from £49, was so high the singer's own website and those of official ticket sellers crashed. The 22 concerts at London's Hammersmith Apollo sold out in less than 15 minutes.
Kate told fans: "I'm completely overwhelmed by the response. Thank you so much to everyone. Looking forward to seeing you all later this year."
Simon Presswell, managing director of Ticketmaster UK, said: "At our peak our website had over 65,000 fans looking for tickets, and our call centre was humming.
"Several thousand fans will be seeing Kate Bush perform live, but despite playing 22 dates a number of fans will be disappointed."
The gigs mark the singer's return to the stage 35 years after her one and only tour and to the same venue, the Apollo, where she effectively retired from live shows after six weeks on the road in 1979.