You can now schedule a post to be published at a particular time and date (in the future) on Medium.
How to Schedule
When you've finished writing your story, click on the "Publish" button in the upper right corner.

The publish dialog will open. Enter up to three tags and then click on "Scheduling / visibility / license":

From here you can also set visibility and licensing for your story, or click on "Scheduling" to schedule your story to be published later.

Click on the radio button marked "Scheduled to be published on" and enter the date and time when you wish to publish:

Note: Your story will be published within five minutes of the time you specify. All publish times are in your local timezone.
Double clicking on the year, month, day, hour, minute, or "AM/PM" will make that field editable. (24 hour time is also accepted)

As soon as you click on "Save" your story status will change to "Scheduled":

Your story will publish at the time specified. You do not need to do anything else. Any edits you make to the story before the publish time and date will be included when it published.
Chrome Browser Only:
You can either type in a new value in each field, or use the small up and down arrows to add or subtract by one increment.

Alternatively, you can access a drop-down date picker by clicking on the downward triangle at right:

Within the date picker, clicking on the month/year name will allow you to select a different month:

And you can also select a different year:

You can also move a month forward or backward by clicking on the small right/left arrows:

Note: You cannot schedule a story to be published in the past.
When you're finished
When you've selected your publication date and time, click "Save":

You will be returned to the main publish dialog, and your story status at top will be marked "Scheduled" and your selected publish date will be listed below.

That's it. You don't need to do anything else to publish. Your story will be published automatically at the chosen time and date.
Notes
- Any changes made to the post before the publish date & time will be published. This means that if the user is mid-edit and the scheduled publish time arrives, the post will be published. Scheduling is not canceled or modified because of edits.
- Scheduling functionality is not yet reflected in the mobile clients. If your story is scheduled to publish, it will be sitting in your drafts like nothing is happening, and then it will go out as scheduled.