Summarize:

It is possible to apply for the card online, or by phone, although online is quicker.  To apply by phone, contact the Apply By Phone Team at 877-621-2639. They are open 24 hours a day. Once on the phone simply answer the questions, and be sure to have your social security number ready. To apply online, follow the remaining steps. Choose the option for the Platinum Card and click "Apply now."  If you have an existing American Express account, sign in. If you are a new customer, choose the option to create a new account. You must complete the American Express Platinum Card application in a single online session. Navigating away from the application or closing your browser will cause the application to time out for security reasons. Make sure to follow instructions carefully and not leave any required spaces blank.  Enter your name as you want it to appear on the charge card. Enter your name, your email and home addresses, your social security number and your date of birth in the designated spaces in the personal data section. Provide any additional information requested on the Platinum Card application. Fill in the employment section with your company's name, city, state and business phone number. Proceed to the financial information section.  Enter your total annual household income and choose the income source from the drop-down menu.  Check the boxes to indicate whether you have a checking account and money market/savings account. Indicate whether you would like to obtain additional American Express Platinum Cards on your account and fill out the information requested for additional cardholders. American Express may notify you of a decision within 60 seconds. You should receive notification by mail within 10 days.
Choose an application method. Open the American Express website to begin your application. Fill in the required information. Verify your information and submit the application.