Adafruit libary and CC3000 shield
I have arrived at a solution that works for me. Instead of using Adafruit library I have adopted Sparkfuns library.

Some adjustments in the setup for the pins:

#define CC3000_INT      3   // Needs to be an interrupt pin (D2/D3)
#define CC3000_EN       5   // Can be any digital pin
#define CC3000_CS       10  // Preferred is pin 10 on Uno
#define SD_CS           4   // Chip select for SD card


Then some adjustemnt in the block for the HTTP request:

client.print("GET /index.html HTTP/1.1\r\n");
client.print("Host: ");
client.print(server);
client.print("\r\n\r\n");
client.println();
Serial.println();


This is not a solution for the Adafruit libary. But it helped me to continue my project.