The problem with references to church fathers is that they get paraphrased from time to time. Sometimes you will just have to look in Augustine for similar terminology and see if you can find the concept alluded to.
That being said this seems the closest citation I've encountered to how you worded it, in Augustine's Marriage and Virginity from the Works of Saint Augustine (Vol. 9 Part 1):
You will not be going to the one who did not dare to raise his eyes to heaven because of the burden of his sinfulness, but to the one who came down from heaven under the weight of his love
Or an alternate rendering:
Thou shall not go unto him, who dared not by reason of the burden of unrighteousness to lift up his eyes to heaven, but unto Him, Who by the weight of charity came down from heaven
You may see the text here.
In general it seems that Augustine in several of his writings associates love with weight:
For the specific gravity of bodies is, as it were, their love, whether
they are carried downwards by their weight, or upwards by their
levity. For the body is borne by its gravity, as the spirit by love,
wherever it is borne.
(City of God 11.28)
There is also a prolonged quote in his book Confessions book 13 chapter 9 where at one point he talks about being carried upward to God by such a weight:
My weight is my love. By it I am carried wherever I am carried. By thy gift, we are enkindled and are carried upward. We burn inwardly and move forward. We ascend thy ladder which is in our heart, and we sing a canticle of degrees...
An attempt to explain this Augustinian concept of weight in simple terms may be found here. For a scholarly source perhaps see "‘Pondus meum amor meus’: The Weight-Metaphor in St. Augustine’s Early Philosophy" which I saw cited in footnotes in a few scholarly works that turned up from searches.
I hope this is what you were looking for.