Here's the song from a source that's usually a very accurate account of the original version of lyrics, Reading Lyrics, by Robert Gottlieb and Robert Kimball (Pantheon Books, 2000), page 448

BABY IT'S COLD OUTSIDE (Frank Loesser)

REFRAIN 1
SHE: I really can't stay!
 HE: But baby it's cold outside!
SHE: I've got to go 'way!
 HE: Baby it's cold outside!
SHE: This evening has been...
 HE: Been hoping that you'd drop in!
SHE: So very nice.
 HE: I'll hold your hands, they're just like ice.
SHE: My mother will start to worry...
 HE: Beautiful, what's your hurry?
SHE: And Father will be pacing the floor...
 HE: Listen to the fireplace roar!
SHE: So really I'd better scurry.
 HE: Beautiful, please don't hurry.
SHE: Well, maybe just a half a drink more.
 HE: Put some records on while I pour.

SHE: The neighbors might think...
 HE: But baby, it's bad out there.
SHE: Say, what's in this drink?
 HE: No cabs to be had out there.
SHE: I wish I knew how...
 HE: Your eyes are like starlight now.
SHE: To break the spell.
 HE: I'll take your hat, your hair looks swell.
SHE: I ought to say, "No, no, no, sir!"
 HE: Mind if I move in closer?
SHE: At least I'm gonna say that I tried.
 HE: What's the sense of hurting my pride?
SHE: I really can't stay...
 HE: Oh, baby don't hold out.
SHE: Ahh, but it's...
 HE: Baby it's
BOTH: Cold outside

REFRAIN 2

SHE: I simply must go!
 HE: Baby, it's cold outside!
SHE: The answer is No!
 HE: But baby, it's cold outside!
SHE: This welcome has been...
 HE: How lucky that you dropped in!
SHE: So nice and warm.
 HE: Look out the window at that storm.
SHE: My sister will be suspicious...
 HE: Gosh, your lips look delicious.
SHE: My brother will be there at the door...
 HE: Waves upon a tropical shore!
SHE: My maiden aunt's mind is vicious...
 HE: Gosh your lips are delicious.
SHE: Well maybe just a cigarette more.
 HE: Never such a blizzard before.

SHE: I've got to go home...
 HE: But baby, you'd freeze out there.
SHE: Say, lend me a comb...
 HE: It's up to your knees out there.
SHE: You've really been grand...
 HE: I thrill when you touch my hand.
SHE: But don't you see...
 HE: How can you do this thing to me?
SHE: There's bound to be talk tomorrow...
 HE: Think of my lifelong sorrow —
SHE: At least there will be plenty implied...
 HE: If you caught pneumonia and died...
SHE: I really can't stay...
 HE: Get over that old doubt...
SHE: Ah, but it's
 HE: Baby it's
Both: Cold outside.

from the 1949 movie, Jupiter's Darling, performed by Esther Williams, Ricardo Montalban, Betty Garrett, and Red Skelton

Betty Garrett turns the tables nicely on Red Skelton in the movie version of the song: https://www.youtube.com/watch?v=7MFJ7ie_yGU

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There are some creepy elements in this song that in these days could result in a sexual assault charge (What's in this drink?). But I first heard the song from Steve Lawrence and Eydie Gormé. And they were married, so I guess it was all right.. Here's a recording:

https://www.youtube.com/watch?v=dImQllSJ36w