The Ghost and Mrs Muir.
It’s not pre-1960, but Murphy’s Romance doesn’t have a false note in it.
It’s not a romance exactly, but The Thin Man is good fun, with snappy dialogue and good acting. It may fit the bill for you because the two leads are a happily married couple who playfully flirt and spar with each other while solving a mystery.
Also, Myrna Loy. Great googily moogily.
You may never have heard of Sabrina. It’s not very famous. But it’s right near the top of my list of the best ever romantic movies.
It’s wunnerful!
So is Roman Holiday with Gregory Peck and Audrey Hepburn.
AAMOF, I would take a look at all of Audrey Hepburn’s movies.
This thread is Freakishly close to home. Its what I got my wife for Xmas.
Except maybe Wait Until Dark- it’s probably the least romantic of her movies.
Yes! One of the most romantic movies ever. And the ending always makes me cry (in a good way).
Very romantic, very funny, and nicely unsappy is Ninotchka. Tagline was “Garbo laughs!” She does, and it’s beautiful.
Here’s a clip. It doesn’t happen until 2 minutes in, but stay with it-- it’s so worth it.
Also, if anyone suggests Magnificent Obsession with Jane Wyman and Rock Hudson, this is another remake of an Irene Dunne movie where the Irene Dunne original is vastly superior.
I am president, vice-president, secretary-treasurer, and greens committee of the Irene Dunne fan club, and I’m here to tell you that anything that is a remake of one of her movies isn’t as good as the original. Unfortunately, the originals aren’t so well known, because the remaking company often bought the original and kept it out of circulation. This is why Irene Dunne isn’t so well known, even though in her own time she was more popular than Katharine Hepburn.
Also, The Thin Man is a great idea.
Does your wife like silent movies? I can suggest some with Mary Pickford and Lillian Gish that are grand, but not everyone likes silents.
Also, after 1960, so we’re safe.
I may pick one silent. Variety should be a bonus.
I want you thank everyone who has responded. My goal will be to choose five, and along with a box of Lake Champlain chocolates I hope to survive another Valentine’s Day. With all these interesting suggestions. I might reprise the idea next year if it all goes over well.
There are at least two versions of Sabrina: The 1954 version with Bogart, Grant, and Hepburn, and the 1990s version with Harrison Ford and Julia Ormond.
I, uh, suggest you buy the former.
If you like musicals, you might want to check out High Society. It’s Cole Porter’s version of The Philadelphia Story with Grace Kelly, Bing Crosby, Frank Sinatra, John Lund, and Celeste Holm.
https://www.youtube.com/watch?v=YG6UllZwj9cWaterloo Bridge with Vivien Leigh and Robert Taylor. A great tragic love story. This one will have her sobbing on your shoulder!
Franco Zefferelli’s Romeo and Juliet, one of the few movies I would call a true work of art.
EDIT: Post-1960, but still… 
If you want to keep with the William Powell theme, I also like My Man Godfrey (the 1936 version). It’s perhaps more fast-talking comedy than pure romance, but Powell does end up with Carole Lombard at the end, whether he knows it or not.
Not exactly slushily romantic, but how about His Girl Friday (Cary Grant and Rosalind Russell)? It’s funny, with a strong and unconventional female lead (and I couldn’t help noticing how, even as the dialogue speeds up as the tension mounts through the film, the actors still manage to articulate every word clearly, unlike more modern Method mumblers).
And why no Fred’n’Ginger? Can’t get more romantic than that.
(PS: Brief Encounter is sad-but-redeemed: remember the last line, where it suddenly becomes clear that the husband knew all about it and was just waiting for her to come back? A different sort of romance).
I agree.
You might also consider Powell & Pressburger’s A Matter of Life and Death if you want something that’s bigger in scope than just the romance. It’s a big beautiful film that deserves more attention.
I haven’t seen the former but I have seen the latter and I, uh, don’t recommend it either (although I like Greg Kinnear in pretty much everything I’ve seen him in).
Silents:
Sad, but with a very happy ending is Way Down East, with Lillian Gish.
Cute, funny, and generally happy is My Best Girl, with Mary Pickford.
Not silent: Another undeservedly obscure film (with pre-fame Lucille Ball) is* Joy of Living*, starring Irene Dunne.
Yeah. So very much. What a waste of talent that useless remake was.
I tried watching that one recently and gave up as the tedious setup kept going and going and going, with no payoff in sight.
I thought I watched it and enjoyed it in the past. So maybe I gave up too quickly or maybe I’m confusing it with another movie.