We need to see pictures of your mower and its engine, to verify that it is as we are visualising it. Most likely the 4 stroke engine is a Kirby Lauson, probably an HK25 (2.5 hp). Most modern 4 stroke horizontal crankshaft engines are interchangeable with the HK25: they have the mounting bolts and the output shaft in the same places as that engine, so they will bolt straight on in place of it. However, the Kirby Lausons had 3/4" output shafts, whereas the equivalent size Briggs and Stratton engine (model 80102) had a 5/8" output shaft, which would involve you in extra hassles to replace the pulley that mounts on the output shaft. So, as Bluegmhtmonaro said in the first place, your easiest solution is to fit a 9 cubic inch horizontal crankshaft 3.5 hp Briggs (model number will begin with 9 instead of 8 - example of a suitable current model is the 93432, but older models are just as suitable) mainly because most of them have a 3/4" output shaft. You could equally easily fit a Honda GX120, an even nicer engine but a more expensive one. Note that where the HK25 was only 2.5 hp, the 9 series Briggs is 3.5 hp and the GX120 Honda is 4 hp - both have plenty of power for the job, and both should be smoother, quieter and easier to start than the Kirby Lauson.
Because both the 9 series Briggs and the Honda GX120 have optional 5/8" output shafts, you need to verify that the one you get has a 3/4" shaft.
Both engines are readily available second-hand, for much less money than a new one. Note that these engines had more than one output shaft length, but one length is usual (approximately 2.5"). Be sure you do not buy an engine with the optional built-in reduction gear between the output shaft and the engine's crankshaft - these are less common than the direct drive engines.
This may be unwelcome and gratuitous advice, but it is quite unusual for Kirby Lauson engines to seize unless they are allowed to run out of oil. 4 stroke mower engines have an oil sump like a car engine, and it is essential that the sump be kept filled to the correct level with clean engine oil (yes, it has to be drained and replaced every year or so). Please do not fit a nice new engine to that nice old mower then seize it up through neglect.
