For those who still fear so much of Russian, Во is used before:
- в, ф + consonant (во Владимире).
- forms of лев, лёд, лён, лоб, ложь, мох, ров, рожь, рот (во льду, во лжи, во рту),
- мне and forms of многие, многое, множество, множественный (во мне, во многом)
- что (во что)
- also во CAN be used with: благо, весь, глава, двор, дворец, дворянство, зло, мгла, мнение, мрак, сколько, слава, сон, спасение, столько, тьма, христос, цвет, человек, чрево (во благо, во весь рост, во сколько, во сне), here sometimes в is more common, sometimes во is, it depends on context.
In this case looking at google в дворце is more used as: в Дворце Спорта, в Дворце Профсоюзов etc. I'm not sure if it's a mistake to use otherwise, it doesn't seem to me.
There are many Russian words where an о or an е can appear or disappear when the word is changed e.g. by conjugating or inflecting it.
For example, звать "to call" but зовут "they call" with an extra -о-; дворец but во дворце where the -е- disappears when the ending is added.
There are historical reasons for the appearance or disappearance of these letters but in the modern language it's probably easiest to just learn the words individually.
A bit like the vowel changes in Latin in some words which English inherited, e.g. efficient versus effective or pronounce versus pronunciation.
"There is no room" is a standard expression indicating just that (i.e. no physical space available). There is no place", on the other hand, is a literal translation from Russian - you will be understood, but you will sound foreign. The problem is that Russian "место" means both a location and available physical space. In English "place" corresponds to the first meaning while "room" (uncountable!) corresponds to the second.
"There is no place" is often used as a part of different common fixed expressions: There is no place like home; there is no place to go.