“To the mind that is still, the whole universe surrenders”: did Laozi say it?
Not in the Dao De Jing
No: “To the mind that is still, the whole universe surrenders” is not in the Dao De Jing; it is Arthur Waley's 1934 rendering of a line from chapter 13 of the Zhuangzi.
To the mind that is still, the whole universe surrenders.
The nearest lines in the Dao De Jing
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不欲以静，
Without desire there is stillness,
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天下将自定。
and the world settles of itself.
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清静为天下正。
Clear and still, one sets the world right.
Where it comes from
The line belongs to the Zhuangzi, the other classic of early Taoism, not to the Dao De Jing. Stefan Stenudd, who keeps a list of sayings wrongly credited to Laozi, found it in the introduction to Arthur Waley's translation of the Dao De Jing, The Way and Its Power (1934), credited to Zhuangzi, chapter 13, with “a mind” and with “still” in quotation marks. The form with “the mind” may come from Huston Smith's The Religions of Man (1958), and the first book Stenudd found giving the line to Laozi is from 2000. We have not seen these pages ourselves.
What the Zhuangzi says
Chapter 13 of the Zhuangzi, 天道 (“The Way of Heaven”), praises the still mind more than once: 圣人之心静乎！天地之鉴也，万物之镜也, “How still is the sage's mind! It is the mirror of heaven and earth, the glass of the myriad things.” Later in the chapter comes 一心定而万物服, “the mind settled, the myriad things submit”, the closest Chinese to Waley's wording. Stenudd compares Waley with the mirror passage; we cannot tell which line Waley meant, but both are Zhuangzi's.
What the Dao De Jing says
The Dao De Jing values stillness too, and says the world settles when one is still, though nothing surrenders. Chapter 37: 不欲以静, “Without desire there is stillness, and the world settles of itself.” Chapter 45: “Clear and still, one sets the world right.”