"The Lady's in Love with You" was written by Burton Lane (music) and Frank Loesser (lyrics) and published in 1939.

The song was a great hit in 1939 when performed by the Glenn Miller Orchestra. On radio it was often listened between June 3 and July 8 1939.

This year of 1939 was indeed very important for Miller's band. They would become forever known with some of their most important standards like "Moonlight Serenade", "In the Mood", "Little Brown Jug" and "Sunrise Serenade".

The film in this clip is part of an advertisement of General Motors to the Chevrolet (Chevy) 1940, that in 1939 was being tested. It focuses the supposedly easy gear shift and was targeted for women drivers to show that driving that car could be as easy as women's make up. A few years later, these same ladies would be making those same cars, Jeeps, tanks, and planes, and assembling bombs to fight against the nazis.

In this music we can hear Miller's voice in a spoken interlude with Tex Beneke.

This interlude of Tex Beneke (TB) and Glenn Miller (GM)) has the lyrics as follows:
TB: (whistling)
GM: Hello there, Texas, whatcha say?
I d like to bend your ear if you're goin my way!
TB: Okay, Glenn, what's worryin you?
Tell me the story and I'll see what I can do...
GM: Well, I got a new girl that s a real killer-diller,
but I'm not so sure that she goes with Mr. Miller...
TB: Hhm, that's nothin to worry about,
Man, don't you know how to go about finding it out?
GM: No, you tell me, Texas!
TB: If there s a gleam in her eye
each time she straightens your tie
You know the lady's in love with you
If she can dress for a date without that waitin you hate
It means the lady's in love with you...
And when your friends ask you over to join the table
But she picks a faraway booth for two,
Well, Sir, here's just how it stands
You ve got romance on your hands
Because the lady's in love with you

Glenn Miller And His Orchestra – My Last Goodbye (side A of the disc)/ The Lady's In Love With You (side B). Label: Bluebird B-10229. Format: Shellac, 10", 78 RPM. Released at US in 1939.

Glenn Miller Orchestra in 1939 included:
Al Mastren, Paul Tanner, Wilbur Schwartz, Stanley Aronson, Hal McIntyre, Tex Beneke, Ernie Caceres, Jimmy Abato, Johnny Austin, Louis Mucci, Bob Price, Charlie Hill, Legh Knowles, Mickey McMicle, Allan Reuss, Arthur Ens, Chummy MacGregor, Bob Spangler, and Rollie Bundock.


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Sources:
Easy Does It - Chevrolet Gear Shift (1940):    • Easy Does It - Chevrolet Gear Shift (...  
Glenn Miller at the Glen Island Casino 1939 (Stereo) :    • Glenn Miller at the Glen Island Casin...  
The Lady's in Love with You: https://en.wikipedia.org/wiki/The_Lad...
Discogs: https://www.discogs.com/Glenn-Miller-...
Glenn Miller Orchestra in 1939: https://en.wikipedia.org/wiki/Glenn_M....
Lyrics: https://www.letras.mus.br/glenn-mille...
Glenn Miller and His Orchestra Top Hits: https://www.colorado.edu/amrc/sites/d...