How can I declare that a text field can only contain an integer?

1. Make your view controller a UITextFieldDelegate by adding UITextFieldDelegate to the class declaration.
2. Add IBOutlets for your text field and your button.
3. In viewDidLoad set your button's isEnabled property to false and set self as the textField.delegate.
4. Implement textField:shouldChangeCharactersInRange:replacementString: method. This method will be called every time your text field is edited. In there, check if the current text field converts to an Int by calling Int(text) and enable/disable your button as desired.

Here is the code:

class ViewController : UIViewController, UITextFieldDelegate {

    @IBOutlet weak var textField: UITextField!
    @IBOutlet weak var button: UIButton!

    override func viewDidLoad() {
        super.viewDidLoad()
        button.isEnabled = false
        textField.delegate = self
        textField.keyboardType = .numberPad
    }

    func textField(textField: UITextField, shouldChangeCharactersInRange range: NSRange, replacementString string: String) -> Bool {
        // Find out what the text field will be after adding the current edit
        let text = (textField.text as NSString).stringByReplacingCharactersInRange(range, withString: string)

        if Int(text) != nil {
            // Text field converted to an Int
            button.isEnabled = true
        } else {
            // Text field is not an Int
            button.isEnabled = false
        }

        // Return true so the text field will be changed
        return true
    }
}

Two things:

1. Specify the keyboard type to only show the numeric keypad. So, set the keyboardType to .numberPad. This, however is not enough to stop the user from entering invalid characters in the text field. For example, the user is still able to paste text or switch keyboards when using an iPad.

2. Specify the text field's delegate and implement shouldChangeCharactersInRange that will not accept any characters other than the digits 0 though 9:

   class ViewController: UIViewController, UITextFieldDelegate {
   
       @IBOutlet weak var textField: UITextField!
   
       override func viewDidLoad() {
           super.viewDidLoad()
   
           // you can set the following two properties for the text field in Interface Builder, if you'd prefer
   
           textField.delegate = self
           textField.keyboardType = .numberPad
       }
   
       func textField(_ textField: UITextField, shouldChangeCharactersIn range: NSRange, replacementString string: String) -> Bool {
           let invalidCharacters = CharacterSet(charactersIn: "0123456789").inverted
           return string.rangeOfCharacter(from: invalidCharacters) == nil
       }
   
       // or, alternatively:
       //
       // func textField(textField: UITextField, shouldChangeCharactersInRange range: NSRange, replacementString string: String) -> Bool {
       //     return string.range(of: "^\\d*$", options: .regularExpression) != nil
       // }
   
   }
   

For Swift 2 rendition, see previous revision of this answer.